This Is NOT A Trick Question. The Famous Snowplow Math Problem

"One day it started snowing in the morning at a heavy and steady rate. A snowplow started out at noon, going 2 miles in the first hour and 1 mile in the second hour. What time did it start snowing?" Say what?! This is a famous problem from the 1942 book "Differential Equations" by Ralph Palmer Agnew. I came across it on Roy Wright's blog (roywright.me/2.... You can assume the snowplow's speed is inversely proportional to the snow that has fallen. And then, using the magic of calculus, we can find an answer! Watch the video for a solution.
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[@SimplyChrist]

It started snowing at 11:59 a.m. At 12:59 p.m., the snowplow driver suddenly discovered he was getting paid by the hour....

[@cisium1184]

The key to all of these "seemingly impossible problems" is always the same: assume something you haven't been given any indication you can assume. In other words, go to the point where critical thought and deductive reasoning diverge, and choose deductive reasoning merely because it enables you to answer the problem.

[@b.f.2461]

Nailed it.

[@rayF4rio]

I think you will find that its true for most higher level math and physics - make assumptions and then see where they lead. Change the assumptions and the answers may change. This is the fundamental problem with modeling (like climate), assumptions are just that...assumptions.

[@kevinm.6015]

and something you have no business assuming. the amount of snow that can fall in an hour doesn't slow down a snowplow, even one that can only travel two miles an hour. especially one that can only travel two miles an hour. BS like this is why people hate math. Good job mate.

[@tianacroaker7310]

@@kevinm.6015 once you reach higher level math classes, the skill being taught is how to make assumptions to create an appropriate model for the appropriate situation. If the situation is “answer this problem on a test before you run out of time” you make simple, not super realistic assumptions that will make the math easier (ie. inverse relationships because they make easier derivations and the plow is obviously slowing at some type of rate). If the situation is “create a model to determine the appropriate time for the highway and transportation department to employ snowplows in the city” you’d probably aim for something more realistic and therefore complicated if you wanna avoid that I-95 36hr gridlock from last winter (at what rate does the snow outpace the plow, at what threshold does the snow height make it hazardous to drive the plow, how long before/after snow starts will civilian drivers be warned off the roads based on the expected rate of snow and how will their driving patterns and traffic affect plow speed and efficiency, etc etc)

[@kevinm.6015]

@@tianacroaker7310 you've convinced me, it's not a BS fake question.

[@MWSin1]

It's a well known fact that snowplows travel at infinite speed on clean roads, and exist simultaneously at all points in space (except those that have snow).

[@peters2928]

guess that's why the plow is on the front of the truck.

[@mayankmotwani2426]

😂

[@jothki]

A slightly better model would assign a fixed but unknown weight to the plow itself, and have the plow's speed be proportional to the sum of the weights of the plow and the snow together.

[@harrymills2770]

Your first statement IS a problem with the inverse proportionality. Pretty awkward how he handled that bit, with the 2nd equation resulting from the 1st, rather than just being the definition of inverse proportionality, i.e. y = k/x for some constant k > 0 is what "y is inversely proportional to x" means. The xy = k is a consequence, afaic, and not the other way around. That's always the problem with inverse proportionality, like the inverse-squared laws for gravity, light and sound. Under the principle, the gravity/light/sound is INFINITE when the distance between two bodies is zero.

[@MWSin1]

@@harrymills2770 In that case, it's only a hypothetical problem. You can't really have zero distance between actual physical objects. If you're close enough for the radius of the object to matter, other effects come into play.

[@002DrEvil]

I reckon in the second hour the driver stopped for a 1/2 hour lunch break, and pretended it had been snowing.

[@jgallantyt]

At first I was thinking the same thing about the inverse relationship between snow and speed, but I discarded it because that's not how snow plows work. They don't approach infinite speed as the depth of snow approaches zero. Also they generally go the same speed until you hit extreme amounts of snow. For those reasons I discarded the idea that it was inversely related and then couldn't solve it.

[@cliffordheseltine6650]

Not to mention that the depth of the snow would not have automatically doubled at exactly the one hour mark. The problem needs to be solved for the entire 2 hour period, not two separate periods since the plow would have been slowing gradually during the entire time as we are informed that the snow accumulated at a steady pace over the two hours. So the speed of the plow would have decreased at an opposite steady pace over the entire period. Breaking the question into two different one-hour periods cannot give the proper answer but calling the local radio station and asking the news desk when it started snowing would resolve the issue forthwith.

[@gulflines1960]

"They don't approach infinite speed as the depth of snow approaches zero."

[@mattislindehag3065]

"They don't approach infinite speed as the depth of snow approaches zero" How can you know that? I live in Sweden and i have never seen a snow plow during the summer. They must be going too fast for my eyes to percieve them.

[@Erkle64]

@@cliffordheseltine6650 That's why all the differentiation and antidifferentiation because he was calculating based on steady snow fall. If he assumed instant snow fall on the hour marks he would have got 11am.

[@scottfree6479]

No, their speed approaches 1 as the depth approaches zero. It reduces from there, but not necessarily to any great extent, hence why k3 is irrelevant.

[@theginginator1488]

The real answer: in the morning

[@lawrenceefting1746]

TheGinginator14 is

[@icecold4411]

Thats legit what I thought he meant by "This is not a trick question" xD

[@hariman7727]

Another good answer: "Wait... is the driver goldbricking, or did we get a blizzard that started a little before noon?"

[@anthonykf99]

pfft, easy. The horses name was Friday.

[@volkerallert5364]

i think this was the name of robinson crusoe´s mate.

[@clefsan]

it was. but is that a reason why the horse couldn't also be named friday?

[@PW-qi1gi]

Wrong. The city has grown in the meantime ;-)

[@brdtrk]

You light the match first!!!

[@tomasgonzalez4819]

You're all wrong! (Simpletons.) The answer, clearly, is "In the morning". :)

[@Dwatthaell]

I never learned calculus, but I do happen to drive a snowplow. I'll freely admit, this math hurts my head. However, from a snowplow operations standpoint, the answer is not what the calculus provides. Even if we assume there's no road pre-treatment occurring, which there would be, but just for simplicity's sake.. The snowplow driver would have been called out a few hours in advance, and would have begun plowing when the snow started to fall. You start plowing a route in those conditions as soon as you can. It saves a small amount of load stress on your plow, and makes the job easier over the long run. *chuckles*

[@92Roar]

I would have absolutely loved that to be someone’s answer cited with “I drive a snow plow I assure you my knowledge in this field exceeds yours”

[@andsalomoni]

The nth demonstration that mathematics has little to do with reality.

[@parkerparker9092]

Touche!!!

[@matroxxm6870]

You sir...or ma'am, have the only correct answer anywhere in this video or its comments section, as near as I can tell...

[@65csx83]

Actually, if we're being literal, this is a 1942 problem. We don't know whether these are urban, suburban, or rural roads. Also, is the snowplow driver a contractor or a civil servant (really important). And another issue about math is that it rarely applies to real life applications.

[@gcree20]

The snow plough's speed decreasing as the amount of snow on the road increases doesn't mean that those 2 quantities are inversely proportional. As many people have pointed out, such a relationship would not satisfy the boundary condition when there is no snow on the road. If the writers of the question expected people to make the assumption that was made in this video then this is a bad question for multiple reasons.

[@aphextwin5712]

To answer the question one has to assume that there is a relationship between snow height and plow speed. That relationship doesn’t have to be linear to (or linear to the inverse of) the snow height. But it can be linear (to the inverse) over a portion of that relationship curve. If we assume that b > 0, we can use this inverse relationship without violating physics. But in the end, you can assume a ton of different relationships that don’t violate physics, the one that has an ‘inverse section’ for part of the curve just lends itself to solve this problem via differential equations. You could as easily assume step functions (ie, speed 1 for height x to y, speed 2 for height y to z, etc.) that make the maths simpler and nobody could accuse of not answering the question correctly. For example, the snow plow starts at speed S when the snow is between zero and x and drops to half that speed when it is between x and 2*x. So, it starts snowing one hour before noon and it takes two hours to reach x. And that is why the snowplough drops its speed by 50% after the first hour is over.

[@mikelm222]

Yeah, for me it's funny that all this complex maths is based on an incorrect assumption - makes my guess of 11.30am after 10 seconds thought seem all the more sensible.

[@JPearlLapis]

I'm gonna go out on a limb here and say the type of person that writes these kinds of questions isn't the kind of person you'd find driving a snow plow... Or any other kind of work truck lol...

[@frognik79]

Now I'm assuming that the snow fall was constant and the plow only went 1 mile in the 2nd hour because it ran out of fuel. You know, since we're assuming things.

[@EaglePicking]

With such an assumption there could be no answer to the question and when there is no answer it's always wrong. It's not a purely mathematical question, but half of a riddle, too.

[@matthewklein660]

We're not assuming things. Everything is based on fact.

[@Jason-it6tp]

@@matthewklein660 The assumption is that the plow's speed decreases as more snow falls. (1:29)

[@BigDBrian]

The assumption should've been part of the problem statement. That way there can be no argument.

[@majermike]

exactly

[@cmc2110]

True

[@deac-hd2lo]

Yeah he completely missed the "in the morning" part of it

[@7636kei]

Also, there's no bloody way someone who have never seen a snowplow (like me, but to be fair I live within the equator belt) would have a flipping idea how does a snowplow work to even make that modelling assumption.

[@EaglePicking]

Yes and no. This is somewhat of a riddle-math-question. You know the riddle has an answer and therefore you know you can make logical assumptions. The only logical assumption possible is the one Presh made. Also: if a student would have assumed that the ratio between plowing speed and snow height was quadratic and would have correctly calculated a result for this assumption, I'm sure the teacher would happily call the answer "correct" :)

[@mindofmadness5593]

Stuff like this is why I became a Photographer.

[@DaRealMcQueen]

We solved it with “assumptions.” That’s some solid maths!

[@Delgen1951]

not really when does morning start? It starts at 12 AM or 00 hours there for 11:23 is wrong. As the snow had (it started in the morning) Started at 12 Am or 00 hours there for had 12 hours to build at a constant and steady rate. This is a badly parsed problem in the morning is a way to lose of a time frame.

[@thecryingshame]

That's all maths

[@gaathabhatia8555]

@@Delgen1951 why does it make a difference when we consider morning starts? In the solution, we are looking for the time difference between 'when it started snowing' and 'noon' i.e. when the snowplow started. Since we got b to be equivalent to 37 minutes, it started snowing 37 minutes before noon. And since any time before noon is technically morning, it still makes sense. Sorry if I worded it incorrectly, English is not my first language. I hope that helps!

[@danly9794]

@@gaathabhatia8555 you are right.

[@Meop79]

Yes, there is no way that snow slows a snowplow at a linear rate. That is ludicrous... so the problem is not solvable because the assumption required to get to this solution is obvious nonsense on its face.

[@EtzEchad]

The assumption that the speed of the snowplow is inversely proportional to the height of the snow is unjustifiable. This problem is unsolvable with the information given.

[@tyronekim3506]

I agree with your assessment that the speed of the snowplow is inversely proportional to the height of the fallen snow requires empirical knowledge about the mechanics of snowplow in operation. That assumption should have been a given or stated as a fact. Many people, including myself, have never operated a snowplow. Also, if the snow level was higher than the snowplow, would the inverse proportionality still hold? This is a bad problem.

[@pearhams2]

An excuse to do calculus; not to solve a real world problem. The math is good but disconnected from the reality with assumptions. Assumed to know the rate of snowfall which wasn't given. Assumed the efficiency of the snowplow was bound to the rate of snowfall.

[@weare2iq376]

Given that the question comes from a book called "Differential Equations", this is actually a fair assumption.

[@tyronekim3506]

@@weare2iq376 It's also possible that the author introduced, discussed, the inverse proportionality in a topic which a student would have read prior to starting the problem.

[@SubduedRadical]

@@weare2iq376 Yes, but that could be true if one assumed, for example, an exponential decay in speed. Which would have changed the answer. For example, instead of assuming x' is linearly related to height, it could be assumed that x' (velocity) is an exponential decay with height. This would change the solution.

[@ddrjay]

TL;DR of the comments section: Too many assumptions, this question sucked and was indeed a "trick question"

[@YASYTU]

TL;DR? Weird way to spell "truth" :)

[@philippenachtergal6077]

Yeah, I guessed what we we supposed to infer but it is NOT a logical inference.

[@VoltzPlayer]

@@philippenachtergal6077 There's literally only 1 assumption and that is that the snowplow's speed is inversely proportional to the height of the snow which is pretty logical

[@philippenachtergal6077]

@@VoltzPlayer No it's not. It is related to it but there is reason to believe that it is linear. I seriously doubt that is as simple a relation in real life.

[@SubduedRadical]

@@VoltzPlayer No, the assumption is (a) that the snowplow's speed is LINEARLY inversely proportional (e.g. as opposed to an exponential decay) to the height of the snow AND (b) that the snowplow has a maximum speed of infinity when there is 0 snow. The second is not expressly stated, but is a requirement of the former. Note the equation x'(t) = k_2 / h(t), which means at height 0, velocity (or x'(t) ) is k_2/0, which is undefined, but in the limit we can see it approaches infinity. This assumption is that the snowplow travels at the speed of infinity on snow-free roads and that its velocity is a linear decay vs height, which is highly unlikely. You can't just look at the first order assumption, but must include the assumptions built into that.

[@Katharsis540]

Was early by 23 minutes. I went by a form of a riddle. 12pm=noon. Then went by the hours and landed at 11am.

[@PrimoStracciatella]

Okay, you really impressed me with all your complicated Math, and juggling all those formulas and equasions were a beautiful thing to watch, but as long as there is no exact information on how much snow is falling per hour, how much snow can be plowed in one hour and how much snow was on the ground, it's simply impossible to come up with an answer - if you don't pull a number or two out of your hat sometime during the exercise.

[@shaggydayshorseshow9567]

I think that "The snowplow's speed is inversely proportional to the height of the snow" was pulled out of a hat - and it's not even a number.

[@cliffordheseltine6650]

I'm not entirely convinced his hat was where he was pulling the missing values from.

[@stevewallgren9175]

Snow plow operators usually do the main roads first, then work on the side streets after the main roads are done. So the first hour could have been a straight line down the main road, and the second hour smaller side roads. These side roads will slow down the plow because of the need for constantly reversing, turning, working around parked cars, etc. In other words, in the real world there is no way to determine the answer with the information given.

[@Xr-pd2oi]

Should you assume feminist or patriarchal snowploughing? I believe feminist snow ploughing does bicycle paths first. www.thelocal.se/20161112/stockholm-transport-heads-defends-gender-equal-snow-clearing

[@jedidiahross7447]

There Sir is a man who has worked before!

[@parkerparker9092]

Good point.

[@BobDaniel]

I got the answer in 8 minutes and 22 seconds, using the "video fast forward" method.

[@martyhorten3743]

I did it in 4 minutes + 28 seconds @ "Speed > 2"

[@amaliatorres5939]

Is this a trick comment?🤔🤔

[@Dubanx]

"Modelling assumption: The snowplow's speed is inversely proportional to the height of the snow that has fallen". Uhm, what? Who reads a statement like this and is like "Oh yeah, that's a completely reasonable assumption"? Does the plow just travel at the speed of light when there's no snow out? Does it sound believable that if a plow can go 50mph in half an inch of snow it has to go 25mph for 1 inch? Does any of that sound reasonable to you? More like "This is a perfectly solvable problem if you're willing to make outlandish and nonsensical assumptions to figure out the answer"

[@BG-mw5pt]

The fact that the snowplow got slower in the second hour is possibly due to how much snow it was falling. So the assumption is reasonable. Also, having no snow it does not mean the snowplow travels at the speed of light, but at its maximum allowed speed. Obviously, it cannot have infinite instantaneous speed.

[@martenkahr3365]

@@BG-mw5pt The math presented in the video does model that the snowplow has infinite speed when there's no snow, and that the plow would move twice as fast with 0.5in of snow as it does with 1in of snow. Which means all the math that follows from that obviously bogus assumption is fruit of a poisoned tree. If there's a maximum allowed speed, it needs to be accounted for. Which as far as I can tell it wasn't during all those differential equations in the middle. The problem might still be solvable, but the way it was modeled here had a serious flaw that makes the solution wrong. Personally, I don't think it's solvable without any additional knowledge about the snowplow's speed and/or performance curve.

[@SmallSpoonBrigade]

Honestly, there's a bunch of assumptions that you have to make in order for this to be solvable. With the number of assumptions involved, it is in fact a trick question.

[@agusrizqi3521]

It started snowing at 6 in the morning. No need magic of calculus, because I can assume it started at 6 in the morning.

[@bipolarminddroppings]

Without doing any heavy mathematics, using my prior knowledge of calculus, I just estimated it would be between 11:20 and 11:30 on the assumption that if it could go 2 miles in the first hour and 1 in the second hour that it can't have started snowing more than 40 minutes before noon. I was pretty happy to have got close just on logic.

[@jollyjoker6340]

Without doing any heavy mathematics, the average snow height over the second hour must be twice that of the first hour, meaning 1:30 is 2x 0:30 and -0:30 is zero. Close, I guess.

[@wiczus6102]

mathematics is logic

[@mickymondo7463]

Likewise, within a few seconds a rough idea of 30 to 40 mins

[@marvinkitfox3386]

Modelling assumption: The base speed of the snowplough, with no snow, is faster than lightspeed. Yup, that's a solid ass-umption you have there.

[@Krasbin]

Probably a better model would be that the speed of the snowplow as a function of snow height h follows the following pattern: v(h) = vmax - k1 x h, where k1 is some drag constant of snow. This models that the snowplow reaches it's maximum speed vmax when there's no snow, and that there is a proportional decrease due to drag of the snow. Since the snowfall is constant, you can say the snow height is constantly increasing over time t at a rate k2, that is: h (t) = k2 x t Plugging this back in into the first equation, you get: v(h(t))=v(t)= vmax - k1 x h(t) = vmax - k1 x k2 x t = vmax - k x t, for k = k1 x k2 is some constant. I forgot the precise numbers in the problem, but I think you can resolve this by taking the change in velocity, the deceleration, between noon and 1 and 1 and 2. These should be, given our previous equation, equal. One can then extrapolate that back until one hits the velocity vmax, as that also means t=0.

[@marvinkitfox3386]

@@Krasbin But then the question would make sense as a Math problem, and would lose its value as an object of scatological ambivalent comparative philosophy. ;-)

[@robertdegroot8302]

@@marvinkitfox3386 An object of what?

[@marvinkitfox3386]

@@robertdegroot8302 of deep thinking about talking shi t

[@alexm.2960]

@@marvinkitfox3386 The problem is that there is a much more reasonable assumption if we want to make assumptions. What you said relates to almost nothing in this topic.

[@JRP357]

If it snowed enough to start plowing in 37 minutes then 2 hours later the plow might not even push it .

[@micheal49]

"If we lose one more engine, we'll never land."

[@revirescomitchell375]

My guess was about 0730, based on the fact that it could only go 2 mile an hour at noon. I just counted backwards in hours until I got to a speed about where I thought the max speed of a snowplow would be, and doubled the speed every time. So it could go 64mph between 7 & 8, 32 between 8 & 9, 16 between 9 & 10, 8 between 10 & 11, 4 between 11 & 12, and the given 2 and 1 mph in the hours mentioned. I based this on real world understanding of the effects snowfall has on roads and driving conditions, and how snowplow operators work. If the depth of the snow also determines the speed, it wouldn't have to double in depth every hours the to cut speed in half, because road conditions don't work like that. Depending on certain conditions, every x inches could double the danger or workload and thus demand half the speed. I call this trucker math. It's based in the real world and you can do it in your head on the fly to give yourself ballpark, neighborhood, or zip code accurate information to help you determine what to do in a given future or ongoing situation. As a trucker if I could only go 2mph, and then 1mph, I wouldn't put myself or anyone else in that situation without a great deal of precautions, like 2 back up plows, a mechanic truck with redundant stress parts for itself, enough food and water for everyone for a week of being stranded, and emergency medical supplies and shelter(s). Plus 2 people in every truck would be cross trained on every truck, and at least 2 workers would be first responder certified, and they can't ride together. THAT'S THE REAL ISSUE AT HAND IN THIS PROBLEM! Blue collar dudes and dudettes are going to get hurt or killed, because politicians are going to make promises or demands that emergency workers butts will be forced to cash. But that's just me being an asshole for comedy sake. I never learned this level of math, I only sat through it to see if my guesstimation was anywhere close. I gotta say, if it started at 11:23, and the plow could only make 2 miles in the first hour, and 1 mile in the second, they must be getting feet per hour, or the plow is moving from a recently effected area to a less recently effected area... regardless, that a butt ton of snow.

[@kfossa344]

Nobody:

[@KrBme78]

It's an exercise in a differential equations textbook, not a real-world solution or an exercise in an engineering. The purpose is not to produce the most accurate model of how snowplows work, it's just to illustrate one way a differential equation can be defined from a physical system to solve a problem without getting overly bogged down in technicalities. The audience for an exercise like this is not people who actually need to guess when snow started based on how fast the snowplows are going, it's students learning math concepts that apply to a broad range of problems (most of which have nothing to do with snow plows) It says "2 miles in hour 1, 1 mile in hour 2" not because that's realistic, but because precise and accurate numbers are not the point of the exercise. The simple numbers make the actual equations easier to solve but don't actually matter in concept--substitute in any more realistic numbers you want and the method of solution would be exactly the same. Similarly, while of course in the real world snow plow speed is not "inversely proportional to the height of snow falling at a constant rate," that setup produces a differential equation that's relatively easy to solve. You could take similar concepts and apply them to cases where the snow does not accumulate at a constant rate, or in which the speed of the plow is affected by other factors, but those equations are significantly more difficult to solve and not reasonable for beginners. Examples and exercises like this are actually very common in intro calculus and differential equations textbooks and classes. DifEQ in particular is all about learning how to relate different rates of change in something in mathematical terms. Obviously that's super important in physics & engineering where assumptions and numbers are more rigorous, but physicists & engineers have to learn the mathematical tools to solve them somewhere. Problems like this (over-simplified representations of physical systems) are common tools because that helps show what the underlying math actually means, which students can then apply to other concepts or more complicated problems.

[@zeos386sx]

In the morning. An imprecise question deserves an imprecise answer.

[@stratocruising]

There is a hidden assumption, that the snow-plowing job is complete after two hours.

[@ankavoskuilen1725]

So a second question would be: When did it stop snowing?

[@321Mdp]

Agreed, or that it is not snowing during the 2 hours of operation, or that you only plow the same road once…. too many assumptions for a real world problem, this is just an attempt to set a maths problem

[@williamervin3272]

I got an estimate of 11:30 when doing it in my head. I assumed the plow would be moving at 2mph at 12:30 and 1mph at 1:30 (the midway points of each hour). If the speed was cut in half between those times, the snow must have doubled. That points to a start time of an hour before 12:30, aka 11:30.

[@UltimaGaina]

The problem is that you are assuming linear dependencies when it's not the case. The speed decrease rate (deceleration) is not constant. Arithmetic and simple geometry are not enough. You need calculus to solve such complex dependencies

[@demiserofd]

No matter what, you're making assumptions. The big difference between this assumption and the assumption made in the video, is that this assumption does not result in the bizarre conclusion that snow plows move at infinite speed when there isn't any snow on the road.

[@reallywilly5052]

It snowed for two hours before it started sticking. So, roughly, 9:30. I was there.

[@gabrieldesmarais7390]

No I live where it snows, one hour after it starts snowing « heavy » all the snowplows are already out and I can assure there is more than 2 inches of snow on the ground after 30 mins

[@skenzyme81]

Love that assumption. 😂 Good thing the plow didn’t head out of the barn right after it started snowing! The sonic boom would have spooked all the spherical cows! 🐮 Heck, if the plow had gone out at nearly the instant the first flake hit the ground, it would have been the first piece of heavy equipment to ever exit its own light cone. 😂 #GoneToPlaid

[@robellett8156]

Sean Kelly You, sir are a genius!

[@BaronVonQuiply]

+1 for the spherical cow reference.

[@MCPhssthpok]

At least the spherical cows aren't in a vacuum, otherwise it couldn't snow.

[@General12th]

+

[@davidporowski9512]

Sean Kelly LOL "spook all the spherical cows" //Are these the targets of cow tipping? That reorient themselves into a grazing, cow-pie making mode of operation ? My best friend had a clown like that which he used as a punching bag! Don't care for clowns much myself, having worked around them at Clyde Beatty Cole Brothers Circus. (I have tales of animal mistreatment that would curl your hair !) One season with them was quite enough, thanks !! WTH?

[@chitranshnigam4796]

And I, 13 year old, thought I'll solve it with play of words while having no idea what calculus is.

[@brendanpmaclean]

It was questions like this that ruined my maths education at school. My first thought was always “Why?” I generally followed that with “Who cares?” My interest in maths was never deep enough to even want to solve the problem. Nobody ever took the time to teach me why (or even if) such problems were important and as a result, I lost interest. Had I been presented with a real world example, I might have seen the point but as it was, all I saw was a problem that had no purpose.

[@airflipper]

The answer is in the morning. It asks what time. Morning in a time of day. The solution is completely based on assumptions. Which is irrational.

[@momzilla9491]

When I was in school the problem was; "What time did the Train leave the station." I always answered; "Before I got there!"

[@andrewtippman]

OMG had to run this through again... This is such an eye-opener and has made me want to return to and brush up on my calculus. So far it is my favourite of your videos - thank you!

[@lllllRBlllll]

“Making this one assumption, we can accurately solve this problem”. *makes a completely false assumption.

[@aniruddhradhakrishnan2471]

Actually think over it, it is not a completely false assumption. Snowplows or for that matter any machine generally has to exert a constant power. That's how mechanical machines are designed. So if we use the fact that P = F v and F is proportional to h(Frictional force), this comes out to be correct. However, it certainly cannot be expected from a math genius to know this much of physics(I am not saying that they wouldn't. I am saying if they don't have, we cant blame them). So this assumption is fairly logical. Of course from another perspective there are a lot of concerns. For example what about the width of snow, temperature gradients etc. But this neat assumption solves a lot of it. Also found your comment two years later, so probably you might not see this lol

[@si-hung9759]

@@aniruddhradhakrishnan2471 But it's also not a completely true assumption either. It's a simplistic assumption whose sole purpose is to find a convenient solution to the problem. For example, if the height of snow is zero, then the assumption means the snowplow travels at speeds that exceed the speed of light. Further, if one came up with a more realistic assumption that took into account average snowplow power ratings, snowpack density, etc. and then solved it, thus resulting in a different answer, then which answer is correct?

[@aniruddhradhakrishnan2471]

@@si-hung9759 Yup this is where practical engineering differs from physics. This is sort of a practical engineering aspect. In a real physical solution like you wanted to obtain we need multiple quantities like variation of snowplow power, friction force, snow deposition rate, snow behavior post falling, heat gradients , apparent weight, etc. Now do you really think that it is correct to ask these questions to a math student whom you want to teach Differential Equations? Also just one final point; as an engineering student I can tell you that if I needed a quick handy solution to this problem, I would use this exact same method. And then think over other factors and try to solve accordingly. Also if there is a different answer, and it is based on more realistic assumptions we will surely take it. This is a good exercise to show that. You know that (1+x)^n ~= 1+ nx for very small x. Try finding (1+x)^8 by first doing 1+8x then (1+4x)^2 and so on for a small value say 10^-5. And try checking the error coefficients. In real life engineering applications we do a similar thing. We dig as deep as we can and then we associate it with saying there is an error factor associated. And while in this example the error factor will be small, in engineering applications generally about 20-25% of error factor is considered. And this method is used widely, unless you are doing extreme precision operations like sending rockets to outer space or building computer CPUs etc

[@mkvk74]

@@aniruddhradhakrishnan2471 You're just replacing the assumption that v is proportional to h with the assumption that F is proportional to h. As you showed these are essentially the same assumption anyway. Both are highly suspect and defending either requires a bit more than rephrasing them.

[@aniruddhradhakrishnan2471]

@@mkvk74 Well in a way the fact that F.v is constant does imply something of that sort right? But yeah you're right about the assumption needing more defence. But then the same thing, the intent of the question matters. The intent here is to teach a simple ODE. Expecting something along the lines of complex machinery equations will be a bit unfair

[@DurangoLegend]

As the snow plow operator, I forgot to mention that the first hour was spent going down hill and the 2nd hour was spent going uphill against the wind. Also the snow fence along the highway influenced the amount of snow on the road. Can you solve the problem now?

[@thekinginyellow1744]

The assumption that the speed of the snowplow is inversely proportional to the depth of the snow is one that only a mathematician could make. No other human beings are so disconnected with reality.

[@trueriver1950]

The consequence of that assumption is that snow ploughs travel instantaneously (ie infinite speed) when all the snow has melted. Look out for the snow plough challenger in the next Grand Prix.

[@madnessbydesign1415]

It's not a 'seemingly impossible' problem, it's an actually impossible one. Assuming that there was no snow to begin with is false logic. The problem states that it started snowing in the morning "at a heavy and steady rate". It could have been snowing lightly and/or intermittently for hours before that. I see so many of these poorly worded problems require the student make baseless assumptions, and that's not good 'logic'... p.s. I'm not even delving into the realities of an official having to call the snowplow operator, the time it takes the snowplow to cover the distance from where it's parked to where it begins work, etc.

[@xnetpc]

It only states that the snowplow started at noon and went 2 miles in the first hour and 1 mile in the second hour. It doesn't say that actually plowed snow during that time, just that it traveled that distance.

[@luckyleafgaming3062]

It also says it started snowing at at a steady rate sometime in the morning, not that such a rate continued indefinitely.

[@KrBme78]

Why wouldn't the roads be clear when the snow starts. Also "it STARTED snowing at a heavy & steady rate" implies it was not snowing at all before, since that's what most people mean when they say it "started snowing."

[@aksukovala181]

@@KrBme78 Doesn't remove the fact that if it already snowed a week ago then they'd use started even if there was already a meter of snow on the ground. Why is this relevant? Because it takes maybe 1-5 times of snowing before the temperatures let it stick, but it can snow for 1-20+ times per winter, meaning that if we're supposed to assume what reality is like with the probabilities and understanding we have, it is far likelier that on a day when it starts to snow, there already was previous snow, unless it was specifically stated that it's the first snow, and using logic, when it's the first snow it's such a bid deal that that's what the people would *actually* say "It was the first snow coming with a heavy and steady rate", unless it had already snowed before.

[@parkerparker9092]

@Lone Wolf Excellent point.

[@johndaddyo444]

The problem is missing at least one more assumption: namely, that there was no snow on the road before snow started falling that morning. Since we assume roads would have been plowed during all previous snow storms it is not a tremendous leap to make such an assumption. There could be another assumption: that the 3 miles being plowed were on flat land. If this were not so, then the problem would need to assert the rate of accumulation of snow was constant over the entire course of the snowplow, not merely that the rate of snow falling was constant. This is because snow does not fall evenly on uneven surfaces due to air currents and eddies. Finally, we must assume there is no melting or compaction of the snow upon falling. This last assumption runs contrary to observation in most of the snowstorms I've experienced over the course of my life. Snow usually falls at temperatures slightly above freezing, which means the ground starts out at a temperature capable of melting the earlier snow which falls. Two things can happen: either the snow melted at the bottom next to the ground refreezes into a sheet of ice, which results in a nonlinear increase in the resistance against being removed by the snowplow, or the snow melted at the bottom runs off the road as water, which reduces the remaining mass of snow & ice which must be removed by the snowplow. And, of course, there's the obvious: snow never actually falls at a constant rate for three hours, and plows do not exhibit inversely proportional rates of removal to depth of snow. The initial removal rates are not governed by the amount of snow, but by the speed limit imposed by traffic safety requirements and local law. Hence the need for the assumption that was given in the problem statement.

[@marksmusiclive]

Amen!

[@nomandatoryvaxing7433]

Haha my Dad walks up and I ask him what’s the answer, he goes “uhhh idk like 11, 11:30” and walks away. Me having no clue how to solve watches the video. Somehow he was correct haha.

[@neutronenstern.]

3:41 so the speed will be infinity if the height is 0 really?

[@R.a.t.t.y]

Assumption 1: The snow plough driver gets a lunch break from 1:30 to 2pm. Assumption 2: The maximum speed of the snow plough is 2mph Thus there is no way you could determine the start of the snowing.

[@BillySugger1965]

Holy crap Presh! This is the difference between mathematicians and engineers. As an engineer, I took the time of the interval mid points for the first and second hours. It took an hour (from 12:30 to 13:30) to double the depth of the snow, so I figured that at a steady rate it took the same time (one hour) to reach the depth at 12:30, so it started snowing at approx. 11:30. So I was 7 minutes out, but it took me 30s and I got damned close and was less time out (7mins) than it took you to describe the mathematical method few of us would have ever got! 😆

[@warrickdawes7900]

But did you add in a safety factor? :)

[@sumner1107]

not engineer vs mathematician that's just a wrong solution to problem with different math

[@hardyworld]

I like the way you think Billy Sugger. Engineers took us to the moon using slide rules to approximate answers to calculations. Understanding the problem enough to determine an approximate answer is often more important than crunching the numbers to find the exact answer. Without this skill, there may be little confidence in whatever answer the mathematician found when he solved the problem. Both problem-solving methods/goals are important to provide confidence in the solution ultimately found.

[@Poultryphile]

And that 7 minutes is WELL within the margin of error considering that I'm still unconvinced that there is enough information in the original problem to say "anything" while sticking with only reasonable assumptions. ~Chemical Engineer

[@obliviouz]

Yeah, this was my method as well - it seemed like much more common sense.

[@myreadingmapped]

What one considers the start of snowing is subjective from person to person. It starts with only one snowflake and grows over time. The fact that it states it was heavy and steady at some point does not necessarily indicate it started heavy at the beginning. It could have taken an hour before it reached a heavy volume. So my guess without all that math was between 10:30 and 11:30 AM. Also the composition of the snow would effect the ability to plow it. wet heavy snow vs, dry light windblown would make a difference and would need to be factored in the calculation..

[@Hypercube9]

Hang on... if the snow plow averaged 2 miles in the first hour (2 mph) and 1 mile in the second hour (1 mph), then the plow would have needed to be going faster at the beginning of each hour and slower towards the end of each hour due to the additional snow that I think we're assuming keeps falling after noon? But these numbers would be correct for the 30 minute and 1.5 hour marks. Between which time 1 hour of snow would have fallen. Consequently, if the plow continues to shovel for a third hour, it WOULD have an average speed of 0 mph over the course of the full hour IF we could have a negative speed for the last half of the hour! But since we can't have a negative speed, we know that the plow would be unable to move after an additional 30 minutes of snow fall! IF there was actually an additional 37 minutes of snowfall before noon, the plow would be unable to move before the end of the second hour! Sorry, I don't know what natural logs are or why you used them, but I have to disagree with your solution. In any case, why would someone need to know what time the snow began? Why WOULDN'T a snow plow driver already KNOW when it began? Why is this plow driving slower than most people walk? How much snow is falling that it can reduce the speed of the plow's on the road by HALF over the course of an hour?!? Why didn't the plow driver start plowing immediately once he saw how much snow was falling? And in what world does snow begin to fall all over, at the same time and at the same rate? Basically, this is a terrible question. And it's probably a big part of why so many people hate math! I mean, if the author was trying to TEACH people how to do math, why would you deliberately choose nonsensical examples with unrealistic assumptions versus real world examples with realistic numbers that people can actually relate to?

[@antivanti]

Snow plow moves at a fixed speed until the height of the snow reaches some limit of what the plow can manage at which point the speed drops off very quickly. Now try to solve it.... Mathematicians should not be trusted with the real world. Mathematicians should invent very handy tools for calculations but leave actual calculations of real world things to physicists and engineers...

[@KrBme78]

Well this is an exercise in a differential equations textbook...for someone (maybe a physicist or engineer?) learning differential equations to apply to their "real world things". Also from the 1940s when pedagogy tended to be much more opaque and unforgiving. The purpose of the text is not to provide the most realistic answer to a physics problem, it's to illustrate the setup and solution of a differential equation that's loosely based on a physical problem one might encounter. This is common in differential equations texts because differential equations commonly arise from physical systems, but the methods to build and solve them are often better illustrated with simplified numbers & assumptions. (also - physicists & engineers make simplifying assumptions all the time)

[@shmajent]

What frustrates me about this problem (and many of these videos) is that the language of the solution makes total sense, is in-line with the mathematical education I have had, and yet I can't see the solution using those tools until explained. It makes sense when you explain it, even the calculus and algebra involved are second nature, but I can never see the solution before you present it. Thank you for posting these videos - even if I'm writing this two years later!

[@darren8453]

If we don't reject b

[@formerjoy7555]

And here I thought the answer was "in the morning" when I saw the thumbnail.

[@burakcanyaman8460]

That is impressive. This reminded me the days I used to take differential equations class in engineering faculty. This course was one of the my favourites! I’ve always been into maths and really like your videos. Keep it up man!

[@csbunbun]

Intellectual 🧐 : ah yes, inverse proportions and calculus My monke brain 🦍 : well definitely not after 11:59 am

[@mikeklass2508]

OR, you could ask the plow driver what time the snow started. I lived in snow country for many years, snow rarely comes down at the same rate over several hours so your problem really is unsolvable.

[@ChannelMath]

"just this one assumption" lol.

[@RedRad1990]

7:50 Ah, yes. Phi, we meet again.

[@parkerparker9092]

Presh, thanks for your riddles. They are all great. Most are way over my head, but I still enjoy them. And good comments and points from all.

[@Jimbaloidatron]

Who cares when it started, when's it going to stop? The snowplow driver isn't going to make it home for tea if this carries on!

[@Wecoc1]

Luckily we can assume that he is more of a coffee man.

[@ssiko52]

I love this channel, it's always snowing in the North.

[@tommybaker4330]

If you can figure out when it started snowing, you can figure out how deep the snow was and find there is no logical reason why it would take an hour to plow miles 2 through 3.

[@shaggydayshorseshow9567]

Union rules.

[@tsisqua]

"We've solved this seemingly impossible problem using calculus." No you haven't. You've made an ass out of you and me.

[@martm216]

Ha-ha - you mean you've ass-u-me(d)! Excellent!

[@tacolands]

Snowplows tend to plow at the same speed regardless of depth.

[@majermike]

@Lone Wolf yea this video is garbage

[@scottrobertson8380]

Agreed. Especially if they're on a major roadway. "Hey Cletus - the snow depth just went from 1 1/2 inches to 3 inches. We better reduce our speed by 50%". Good grief................smh..................

[@hungryjack8032]

Since when does any city employee start working in the middle of their two hour lunch break?

[@ajaded1]

A part of me used to think that I should pay attention to these equations. 30 years later, neither side has budged.

[@NotYourAverageNothing]

I’m sorry, but you’d have a hard time convincing me that there is enough information. We don’t know how much of the snow the plow plowed, or whether it ever stopped snowing. Also, shouldn’t it be h(b), not h(-b)?

[@MrHatoi]

We know it didn't stop snowing because it said the rate is steady. If it stopped snowing, the rate would change to 0. How much snow the plow plowed is completely irrelevant, we know that the overall speed of the plow is inversely proportional to the current height of the snow and that's really all we need. Also, it's h(-b) because the parameter for h is t, the number of hours after noon, while b represents an amount in hours before noon.

[@NotYourAverageNothing]

MrHatoi You just confused me more, especially with that last part. You just stated the very reason why I think it should be h(b).

[@MrHatoi]

Ok, let's go piece by piece here. We don't need to know how much snow the snow plowed. It's not relevant to our calculations. All we care about is how fast it goes, which is information we do have. It's not a realistic measurement by any means but it's given as a "modelling assumption" in the original problem, so as far as we're concerned it has to be true. The problem states that the rate of the snow is steady, i.e. constant. The rate of snow is the same as the number of inches added each hour. If the snow stopped, the rate would become 0, i.e. 0 inches of snow are being added each hour. However, that would require the rate to change from whatever it was before to 0, which would violate the condition that the rate cannot change. Therefore, at least within the timeframe of the problem, it doesn't stop snowing. Let me highlight this in bold because I think you missed the key part of why it's h(-b). t represents time *after* noon while b represents time *before* noon, so in order to substitute b for t, we have to take its opposite to represent time after noon. For example: If it started 5 hours before noon it's the same as if it started -5 hours after noon.

[@NotYourAverageNothing]

MrHatoi But we’re trying to represent time before noon. “At b hours before noon, the height is 0.”

[@MrHatoi]

OK, drill this into your head: *b represents time using hours before noon* *t represents time using hours after noon* *Those are two different units of time, so if we want to substitute b for t in the expression h(t), we have to take the opposite of b to convert it from one to the other.* It's exactly the same reason why you can't plug a measurement in inches into a formula that works for meters and expect to get an accurate answer.

[@eightbars1]

I live where there are more than four different types of snow plows. Each can move different amounts of snow per hour. Which plow are we using here? The type that is on the front of a truck? The type that is on a tractor? A bull dozer? A road grader? A 1 1/2 yard dump truck? A 3 yard dump? Each of these move different amounts of snow and can handle different amounts of snow per hour. And one of them slows down to handle corners neatly. Too much being assumed here.

[@cdmcfall]

I assumed that since the rate of snowfall is steady, the depth of snow over time would be a linear function with an x-intercept at some value of -t. Because of that, I assumed that the deceleration rate of the snowplow was constant (ignoring the initial acceleration to get to its maximum velocity), so I was trying to figure out that deceleration rate, which would just be some f(x) = -x^2 type of thing with it's maximum at t=0 (noon). From there I was hoping to find some intersection between the linear function and the quadratic at some -t value, but I couldn't figure out how to derive those functions. Doesn't seem like I was on the right track anyway. This is a very non-intuitive problem, but I love the logic. I'm also happy it involved the golden ratio!

[@klardfarkus3891]

The problem seems to assume that the speed of the snow plow varies with the depth of the snow. That is not true.

[@321Mdp]

Agreed, the efficiency of the plow is important to calculating the answer. He just assumes the efficiency is inversely proportional to the amount of snow…. So as the amount of snow approaches zero, the speed of the plow would reach towards the speed of light

[@RealRuralJapan]

Snow melts faster after noon when it’s hotter also. Snow never settles straightaway when it snows either as it needs time to cool the earth before it can do so without melting. Also the temp of the earth when it started snowing will effect when the snow can eventually settle. Snow can fall for hours before it’s cold enough to settle. As a logic problem it falls very short.

[@ptviwatcher]

The average speed in the first hour is twice of the second. Hence the average snow hight of the second hour (at 13:30) is twice that if the first (at 12:30) This means it started snowing at 11:30.

[@jessstuart7495]

I basically did the same thing, but actually wrote out the snow-accumulation function and solved for the start time of 11:30. I calculated the average snow accumulation from the endpoints instead of realizing the average values occur on the half hours. Very nice!

[@akaRicoSanchez]

I had a similar idea and reached the same answer, but then I realized it is wrong. If the assumption is that the snowplow goes slower the higher the snow, you cannot work in average like this. The snow piles up at a constant rate, but, during the hour, the snowplow goes slower and slower, so if you plot the height of the snow in function of the position of the snowplow, you get a curve and your method (and mine!) approximates this curve as two segments. That does not work :)

[@quaironnailo]

The height function could be a sin^2 wave for all the snowplower cares. All we know is it moves as much snow in the first 2 miles as in the last 1. The width of the snowplower is constant, so as long as the average height on the second hour is equal to twice the average height on the first, this requirement is fullfilled. It doesn't matter if it's a curve, a slope, or just two giant blocks of height h and 2h: as long as the average is h and 2h, the snowplower will go half the distance in the second hour than in the first. So if we assume the snow falls at constant speed K, and t the time the snowplower started working (being 0 the time when it started snowing), the equation is 2(Kt+K(t+1))/2=(K(t+1)+K(t+2))/2 We can multiply both sides by 2K, so it becomes 2(t+t+1)=t+1+t+2--->4t+2=2t+3--->2t=1--->t=1/2. So it started snowing half an hour before the snowplower got to work, so at 11:30 AM If this is wrong, i sure as hell can't understand why.

[@ptviwatcher]

That is true for position, but not for speed, which decreases linearly with the height of the snow. So, if speed decreases linearly, then the average speed within the hour occurs at half hour. If the question was "where was the snowplow when it started working" that would be a different problem, but since the question is "when" I can still see no problem with this analysis.

[@ptviwatcher]

But since the problem was in a differential equations book, if my life depended on it, I'd probably use Presh's answer :) I think he should use one of his famous graphical answers to make it clearer!

[@rebelsilk2016]

SO MANY COMMENTS attacking the assumption he made. The assumption is solid, but somewhat poorly phrased. So what do we know? We know that it snows at a HEAVY and STEADY rate - that information is in the question. We also know that there's quite a bit of snow, both since the question specifies HEAVY snow, and the fact that the plow only manages 3 miles in 2 hours. We also know that the snowplow slows down drastically in the second hour compared to the first hour. Since we're not given any information about what might have caused this other than that there's more snow in hour 2, it's fair to assume that FOR THE AMOUNT OF SNOW WE'RE DEALING WITH, the speed of the plow is inversely proportional to the height of the snow. That means that a more accurate way to phrase the assumption would be that the snowplow's speed is inversely proportional to the height of snow that has fallen, FOR VALUES OF SNOW HEIGHT, or h(t), ABOVE a given value. That would eliminate the whole "with no snow the speed is infinite"-nonsense. Now, with the information we've been given, we can clearly infer that the height of the snow is above the value needed for the assumption to be correct for the entirety of the time the snowplow's been operating. Thus, the assumption is solid.

[@popogast]

MindYour Decisions I suggest a more "geometric" solution. If the snow fall started at 11:30, the heigt of the snow would be double at 13:30 than 12:30. This would correspond to the average speed of the snowplow.

[@pvanukoff]

Exactly what I thought. This feels more right to me than his answer.

[@kevinwilliams510]

Fortunately for everyone on earth who at some point in their lives relied on the solution to a differential equation (and that would be just about everybody), feelings aren't involved in the calculations. If I have to average 60 miles per hour going up and down a hill, and I average 30 mph going up, how fast to I have to travel coming down? It "feels like" the answer should be 90 mph, but that is WAY off.

[@pvanukoff]

Sure, I didn't say it was right, just that it felt right. 25 years ago, back in high school, this type of problem would be a breeze for me, but it's been a loooong time since then :) Presh's analysis didn't click with me. A graphical analysis would go a long way towards proving his point. I'm not the only person suggesting that 11:30 is the answer to the problem, so apparently, Presh didn't explain it well enough.

[@kevinwilliams510]

The problem with this reasoning is that the speed of the snowplow isn't increasing or decreasing at a constant rate, so average speed over any interval cannot be extrapolated backward or forward to give you the correct answer. In this case, it gives you a reasonable approximation, but not the correct answer.

[@marfanity]

In Kevin Williams's example, going up a hill at 30 mph and down at 90 mph, I can see how that wouldn't average to 60 mph, assuming the distance up is equal to the distance down. However, if instead we were driving 30 mph for a half hour and then driving 90 mph for a half hour, (blocks of time instead of distance) then we would have driven 15 + 45 = 60 miles after one hour for an average of 60 mph. The problem states that the plow drove 2 miles in the first hour and 1 mile in the second hour. So the average speed driven in the first hour was 2 mph and the average speed driven in the second hour was 1 mph. That sounds more like my time example than Kevin's distance example. The average speed is twice as fast for the first hour than it is for the second hour. So if we say the average snow height during the first hour is 1 unit, then the average snow height during the second hour must be 2 units. Since our plow starts driving at 12, and the snow's falling at a steady rate, wouldn't that put the 1 unit mark right at 12:30 and 2 units at 1:30? So at 2 pm, we'd have 2.5 units of snow, and at 11:30 am, we'd have 0 units. Where did I go wrong?

[@deeptangshunathnath4668]

When teachers don't want u to pass the examination at any cost of the bitcoin...

[@chriswright9096]

I got a different answer but one that deserves some attention. I did not assume that speed is inversely proportional to height of snow. Instead I assumed that the amount of snow cleared in an hour is a constant (therefore it takes longer to clear deeper snow, but the amount of snow cleared per hour is consistent). So, in the second hour the average snow depth was twice that in the first hour. Lets say that in the first hour the snow went from 0.5ft to 1.5ft (averaging 1 ft). In the second hour the snow went from 1.5ft to 2.5ft (averaging 2ft). If we follow the straight line backwards, we see the correct answer is 11:30am (the point that snow height was zero). I know it all depends upon the initial assumption.

[@raulcarlos2937]

What a lie! There is no way the snow plow would start at noon! At lunch time? City, County, State, and Federal employees all go to lunch at 12:00, stop working about 10 minutes before, clock out, and eat! At 11:23 they would have been driving back to the yard, After eating they would return to clock in at 1:03, then put there lunch boxes away, then walk over to the truck, kick the tires, turn on the truck, warm it up, then start plowing about 1:45, maybe, but its possible it was a private snow plow company getting prevailing wages then I would believe it started at 12:00 noon! Lol

[@danpowell806]

Which is why the plow only moved two miles from noon to 1. The driver forgot the parking brake on the hill, and sent the plow into a ditch. The wrecker arrived at 1350 and pulled it a mile to the repair lot.

[@visaman]

Unless it is a union job and they get paid by the hour, so they would be on the road 24 hours before the snow falls.

[@paperEATER101]

true ...but things were different in 1942

[@saraflint2982]

*their. Come back when you can spell above a first-grade level.

[@xnetpc]

The 2018 way to solve this to use Common Core methodology. How many extra steps would Common Core methodology add to solving this problem?

[@stevenmellemans7215]

Snowplow on steroids. I'd like to race with it when it isn't snowing :-)

[@shtfeu]

Steven Mellemans Odds that the driver was Homer Simpson are infinitely small.

[@Lightning-Shock]

Trust me, you don't. According to the assumption, this plow would go at an infinite speed when it doesn't snow. Not only you wouldn't stand a chance in theory, but practically that would instantly create a huge black hole what would swallow the entire Earth and not only.

[@Pecuniarly]

I made the assumption that the snow in the second hour was twice as high as in the first hour. Then I did a simple approximation, on purpose without units: first hour snow height averages to 1, second hour it averages to 2. To get these averages with a steady height increase, the snow height must go from 0.5 to 1.5 in the first hour and from 1.5 to 2.5 in the second hour. So height increases with 1/hour and is 0.5 at noon, so it started around 11:30. Then I watched the video. Turned out to be quite a decent approximation, for a one minute analysis by head.

[@aryanshrajsaxena6961]

Can't we assume a linear decreasing relationship between height of snow and speed of the plough, with a finite speed of plough in the absence of any snow? because in this case, plough will have infinite speed when h=0

[@richdobbs6595]

Typical mathematician to use an assumption that implies snow plow is going at infinite speed at the start of the problem!

[@warrickdawes7900]

Well, technically it would only be going at infinite speed when there is zero snow, but it did not start plowing until after some snow was built up. But yeah, infinite things are stock and trade for mathematicians I guess.

[@badrunna-im]

Rich Dobbs The speed will only be infinite when there is no snow in which the snowplow will be irrelevant anyway, so it's safely out of the domain.

[@richdobbs6595]

Badrunnaim Al-Faraby - Good point. The assumption just implies that an infinite speed if they start up the plow before the snow.

[@danpowell806]

Or if the snow melts before the plow ceases to exist.

[@hj8607]

Safe to assume snow may even have started very early and heavy . By noon 6' deep and by 1 PM 7 ' deep . (no infinite speed is implied. Truck always faces resistance even with NO snow) Simple linear equations requiring simple algebraic (given a bit more data) solving and using calculus just shows off what he doesn't know about the subject (calculus). (I endured 4.5 years of the subject )

[@theodorostsilikis4025]

This is not necessarily a calculus problem, it can be solved using similar triangles. Base of an orthogonal triangle is time, height is the height of snow, and by choosing two points at the base for 0 hour (when the snowplow started moving) and 1 hour (the 2 hours point is the end of the base,where the right angle is) you get 2 trapezoids. The big trapezoid has double the area of the small one as defined by the problem. So the question becomes what is the length of the base (spoiler* Base=2+φ)

[@jaynecobb3701]

That one is really simple. They just give you the answer right up front. What time did it start snowing? It started snowing in the morning. Don't overthink it.

[@MCPhssthpok]

I hope this question came with instructions to make whatever assumptions you think you need and was marked on technique rather than on the final answer. Otherwise it is just pathetic. As other people have pointed out, the assumption made is absurdly unrealistic and different assumptions would give a different answer. Asking us if we got the answer in this case is basically asking if we guessed right.

[@hariman7727]

I was off by about 23 minutes, compared to the answer in the video. 11 am seemed like the proper start time for snowfall because of the rapid slowing of the plow and other factors.

[@_FirstLast_]

I love how everyone is missing the point of all this. Sure, this problem uses variables based on assumptions (that's not in dispute) -- the equations are good, though, and the math is solid. The framework of the equations are what's important, not some arbitrary or assumed value that will be plugged in after-the-fact. Many values are unknown and will absolutely change the final answer, but the path to said answer will not change -- the framework, the math -- that is what's important. And the foundation is solid. In exactly the same way the quadratic equation works independently of what variables you plug into it. The framework (equations) will produce a solution relative to the variables used (assumed or otherwise). If the assumptions used to find said solution are indeed true in practice, then your answer will reflect that truth. If the assumptions are wrong, then the answer will reflect a "correct" wrong solution. This is a lesson in thinking outside the values and focusing on the mechanics of the equations....when you may not have the actual values to use for your variables (as happens often in the real world). Change variables about snowfall, traction, horsepower, visibility, temperature, dead battery, union breaks, or name-a-thing and the outcome will change --- but the MATH does not change. The math is the tool to reach the solution and cares not for what values are plugged in (at least in this example).

[@geoffroberts1126]

'Variables based on assumptions' Translation: Wild guesses. And nothing in that guesswork gives any form of validity to any answer other than that which is supplied in the question. "In the morning." That's literally the only answer you can arrive at from the supplied data. Anything else is a guess and can neither be confirmed nor disproven based on the information provided.

[@pikomonde3933]

As soon the hint is given, I have a hunch that the answer is between 11.00 and 11.30-ish , but i'm too lazy too count it using calculus. Edit: how i have that hunch? I have a hunch that the answer is noon - 1/2 hour - some minutes (might be in 1/4+1/8+1/16... series)

[@BrittanyArtPoetry]

When I first heard this problem my initial assumption was that after 2hours he had to turn back and redo the first mile since it had continued to snow and undo his work

[@quixotix9540]

It started in the morning...

[@HenkJanBakker]

There is only one fact to be derived from the info. The snowplow is slowing down. And as the snow is constant I will assume the deceleration is a constant too. If you can not state that this whole problem is unsolvable..... And as that is not how snowplows work, this is just a theoretical logic puzzle. And due to lack of numbers, not really a math problem. Anyways: Introducing the term "miles of snow per hour" The more snow, the slower the plow gets. But as we don't really know (or care) about the speed/deceleration we focus on the miles traveled through snow per hour. If the distance traveled is reduced by one mile and the first distance was 2 hour trip we could deduce the second hour also reduced the speed by 1 mile per hour so the the snowplow came to a full stop. So the snowplow was forced to stop after two hours and both hours cut the ability by a mile. During the activity "3 miles of snow" fell to make the snowplow stop. So about "a mile of snow" fell before the plow started. Guestimating I would say; average of "3 miles of snow" in 2 hours. So a little under 40 minutes. Makes it a little after 11.20 AM

[@williamjones9395]

I was never good at higher math. Only went as far as algebra in high school, and did fairly well. But did not go on to higher math as I had no inteterest. Learned some geometry along the way. Enough to get through life, and I've done fine. But these equations are so involved, they seem irrational because there alot of assumptions made or things not accounted for in the real world. Such as, how deep was the snow, how fast was the plow going? Did the plow slow down because of the depth of the snow increasing, or was the driver being careful. And in real world circumstances, there is no possible way you could truly determine when that snow began. You measure its depth over time if it was falling at a steady rate. But would the snow have been deep enough in such a short time to deploy the plow? And if you started plowing at noon, you had to have a darn good idea when it started because you had to start plowing. These mental exercises are challenging for those of us not well versed in higher math, and it's good to work that grey matter. But in the real world, this means little if anything, and has little to do with real life.

[@catman64k]

your asumption is false. A snowplow has a top speed and also a limit where it can work. maybe the topspeed is sth like 80km/h or 50mph. But we also have to consider that we are in the year 1942. so maybe i should decrease the possible topseed and assume sth like 25km/h or 15mph also at certain snowheight even a snowplow gets stuck. So it will move 0km/h / mph The next problem is that the moving speed is also not inversely proportional. Maybe its more inversely quadratic proportional. okay, now the next problem, which makes the question very intersting: is the snowplow moving the same route in the 2nd hour. If so, we have to reduce the snowheight to zero after the snowplow has passed. the next thing is, that we have to consider that the snowplow will lose speed within the hours, as the snowheight further increase. The solution is: collect more information

[@ianmoseley9910]

catman64k - in the UK early 1940s you would have to make allowance for bomb damage from the previous night's air raids

[@susananastasiadis3923]

It says it started in the morning though.

[@MrArtVendelay]

An empty parking lot is covered in snow. The snow plow comes along and at some point, half the lot is done and half remains. The plowing of the remaining half is continues and at some point half of that portion remains unpaid. Well as you continue plowing at some point you will have half of what is being plowed always left and you can halve the remaining portion of the lot an infinite number of times. Therefore, you can never finish plowing the lot.

[@paygemitchell5330]

There is no way to determine if the the speed of the snow plow is decreased by half every hour, or if it was decreasing by one every hour. Also, there is no way to figure out the base speed of the snow plow when there is no snow, as many people have pointed out, based on this solution, the base speed would be infinite. Lastly, It's really the volume of snow that would matter (which is cubic, not linear) that would matter in house fast a snow plow would be able to go. There's simple not a real world solution to this problem with the information provided. Side note: as someone who grew up in a heavy snow fall area, I would love to see the amount of snow required to make a snow plow that slow. Under normal circumstances snow plows go at least 5-10 mph in residential areas., and it doesn't really matter how much snow there is. The amount of snow needed for this problem would be measured in feet, not inches.

[@Alan-jk1yi]

This is similar to Fermi's "How many piano tuners are there in the city of Chicago?" question, in that it's not asked to get a correct answer, it's asked to get the person thinking creatively to try to reach a reasonable answer even though it initially seems impossible. A mathematically minded person would say that neither can be answered correctly with the given information because there are far too many variables. And they are right, but that's not the point. The point is to make a good faith mental effort rather than just giving up before you try and just say "it's impossible". Fermi came up with what was at least a non crazy answer to his question, and this was at least a non crazy answer to the snow plough question. You may disagree with the assumptions made, but hey, at least their brain was working, and they actually gave an answer. We never make progress on anything if we don't try.

[@cameraredeye3115]

Damn, it's been a while since I got my feet wet with problems like these...

[@tlawhon]

Too many assumptions: the snow stopped, the snowplow started after it stopped, snowplows go at a steady speed...

[@earlsreid4130]

I live in the south, have never seen a snow plow. It doesn't snow very often here.

[@lawrencefranck9417]

You answered the time it needed plowing not what time it began snowing.

[@Davidjune1970]

The issue with the problem is that it snowed heavy and steady in the morning. It does not say that the rate also applies to the afternoon. Now does it allow for the possibility that the plows speed is affected by other variables like traffic that is being bogged down by chaotic driving.

[@ozzygilliam9194]

11:30AM if you don’t assume, if they plowed a mile per hour of snowfall, then in the first hour they plowed twice the distance than in the second, which means snow fell for half as long. Which is 30 minutes, equating to 11:30AM.