Calculus (Basic) WORD PROBLEM Why Calculus is so POWERFUL!

[@charlesmitchell5841]

No need to apologize for the length of the problem. Calculus is the real deal. Thanks for the lesson.

[@bandini22221]

This takes me back to my first Calculus course, in the 70s, finding minimums and maximums. I've forgot almost everything.

[@sarathkumaraprematilake6067]

Me too.............

[@JayPatel-fo4ss]

The easiest way to solve the problem is to add heights of both the poles (12+28=40) and then find the proportion of each pole against this figure (smaller pole represents 12/40 or 30%). That is the ratio of the position of x i.e. 30% of 30 is 9ft.

[@dabneyoffermein595]

how did you do that? omg

[@billblumke3506]

About some of the comments: Of course there are other ways to solve this problem. However, the point is to learn and practice using calculus. For that purpose, this is an excellent presentation!

[@dabneyoffermein595]

if you like root canals yep!

[@benavidezf]

Congratulations, you just proved that the shortest distance between two points is a straight line. You can get the same result by projecting the 12 foot pole into the ground and drawing a straight line between the tips of both poles. This gives you 12 + 28 = 40 on the vertical side and 30 on the horizontal which is a 3, 4, 5 triangle. The ratios of 28 and 40 is 7/10 so 30 x .7 is 21. Or 21 feet from the 28 foot pole and 9 feet from the 12 foot pole.

[@thomasabuckley2188]

Flip LH Δ down on Hor axis. Shortest distance between pole tops = straight line. Inspection shows angles same at ground point. Tans same = 12/x = 28/(30-x). Solving x= 9. No calculus!

[@richardhole8429]

Dang! Wish I had seen that solution. Your algebra beats the calculus!

[@richardhole8429]

Everybody, do scroll down to @thomasabuckey2188 for a very simple solution using algebra. Solves this in seconds. Imagine moving the right pole downward from the x axis. For all values of x, the wire length is the same. Draw a straight line between the posts to get the shortest wire. Brilliant solution, Thomas!

[@black__panther108]

can you explain more

[@thomasabuckley2188]

⁠@@black__panther108 OK. Here is more detail. Draw left pole down from ground level. The top of this pole, now pointing downwards, is the same distance from ground level. Now looking at pole tops, the shortest distance between them is a straight line. Where this direct line crosses the ground line is the required point. Notice that opposite angles, where these straight lines cross are equal. So the Tangents are the same. Left Δ, Tan = Opposite/ Adjacent = 12/X. (Using same references as used in calculus form of solution) Right Δ, Tan = 28/(30-X) So 12/X=28/(30-X) 360 - 12Z = 28X 360 = 40X X = 360/40 = 9. Note this method confirms that the extensive calculus solution provided is without error.

[@thomasabuckley2188]

Typo error: 360 = 28Z + 12X, should be 360 = 28X + 12X. Did not affect calculation.

[@scottneighbors7077]

Just draw a straight line from the top of the 12’ pole to the base of the 28’ pole and a straight line from the top of the 28’ pole to the base of the 12’ pole. At the point where the two intersect, draw a line perpendicular to the 30’ base that passes through the intersecting point. Basic map orientation….shortest distance between two points is a straight line. In this case we have two short lines that cross.

[@riziqabulawi9627]

I thank you for this opportunity to watch and help

[@oladelebamgboye2813]

Fabulous explanation takes me back, thank you!

[@Tim-Kaa]

Thank you

[@DaveRissik]

It is interesting to note that the solution results in a common ratio of 4/3 (pole height to distance of anchorage from their respective bases) of the two triangles. This makes sense since such ratio leads also to minimising simultaneously their hypotenuse lengths which is what was set out to be determined!

[@Kleermaker1000]

Indeed. This means that there is also another, much easier way to solve this problem: find the same proportions of both right triangles: 12/x = 28/y, the sum of x and y must of course be 30, so y must be 30 - x. Then 12/x = 28/(30 - x) => 360 - 12x = 28 x => 360 = 40 x => x = 360/40 = 9. :)

[@bizikimiz6003]

Precalc solution is to flip the triangle on the right side down. OK, so the shortest is the straight line. The line if the left pole's bottom is in the origin, y=-4x/3+12, which gives us, for the connection with the ground, x=9.

[@JayAlfredoG]

Makes you appreciate Hidden Figures more. It took all this grinding at math to get one simple answer about a string I’m and two poles. Imagine what it took to work out angular velocity and momentum while dealing with gravitational pull from two bodies.

[@godwinkoroma9024]

Superb Explanation!

[@tarekraja7537]

Excellent programme and analysis. Thank you sir and salute to you sir

[@johnmaynor3851]

Imagine the amount of paper to solve this through just algebra and geometry, calculus is a way of saving the trees.

[@stub1116]

I really enjoyed his video.

[@bpark10001]

You don't need calculus for this problem, only geometry & algebra. Flip one of the poles over, so it pokes down into the ground. Draw straight line to connect the poles. This line will cross the ground line. That's the point for the ground stake. You have 2 similar right tringles, in ratio of 12:28. 12X + 28X = 30. 40X = 30, X = 3/4. Distance from 12' pole = (12)(3/4) = 9'. Distance from 28' pole = (28)(3/4) = 21'. A better problem is to determine the ratio of height to diameter of soup can to determine the most volume to surface area, or how much beer to drink from a can to maximize stability (lowest center-of-mass).

[@chrisboonzaier7813]

Yes you can, but the whole purpose of this exercise is to practice and learn calculus.

[@winterstar4813]

i love these

[@sprky777]

I did a quick estimate by taking (h1/(h1+h2))×d resulting in 9.

[@TexasElectrician77]

Thanks for the instruction! True story, my limited working knowledge of algebra but a little more of geometry led me to the correct answer of 50’ by estimating the distance at a 2-1 ratio and then calculating the length of the respective hypotenuses. I would like to learn more advanced math because my high school course in introduction algebra was not very productive. My tendency was to come up with the correct solution but not be able to show all the proper steps and thus be penalized as if I got the answer wrong. The process of working through the problems didn’t make much sense at the time, I understand that it’s fundamental to being able to work through more complex problems later on. I never had a geometry class but somehow picked up a lot of basics like angles of a triangle, degrees in a circle, the Pythagorean theorem, etc.. What are the defining characteristics of calculus as a mathematical system? I have seen graphs and vectors plotted but don’t have a lot of knowledge about it in general. Thanks

[@ve2um]

17:55 Debut of the calculus process.

[@jameslof2451]

Can also think of shining light from the top on the one pole to the ground which acts as a mirror. Light will take the shortest path. Angle of incidence equals angle of reflection, creating 2 similar triangles. Simply solve as a ratio problem. Also is you look in the mirror and see one pole inside the mirror, you will have a large right triangle with base of 30 and height of 40. Therefore, the shortest distance will be 50.

[@gpil2000]

Jeepers Batman! LOL I will replay at least 5 more times. Thanks for the exercises.

[@adriancazan6957]

I think the problem may be solved by geometry/trigonometry by reflecting one pole against the horizontal line and joining the end of the other unchanged pole with the end of the reflected pole. From engineering point of view consider the horizontal line as a mirror , but reflecting only one pole. The length of the straight joining line is the shortest length of the cable and one can also calculate where the connecting line crosses the horizontal line (between the poles).

[@KenFullman]

I do a similar thing when playing snooker. If I'm going to need to rely on a bounce off the cushion, I imagine the cushion as a mirror. I then aim for where I want the ball to go on the apparent table in the reflection.

[@gregorymagery8637]

Another way to solve the problem : If you turn the second pole downwards, the shortest path between the ends of the poles is a straight line. This creates 2 triangles of the same shape of different sizes. => x/12 = (30-x)/28 |______ 28x = 12*30 - 12x | 40x = 12/30 | x = 9

[@gregorymagery8637]

Typo -3. line should be 40x = 12*30

[@jigneshsoni9263]

@@gregorymagery8637 how did you conclude it forms a similar triangles? I understand it would look similar visually. I am wondering what is the proof?

[@gregorymagery8637]

.@jigneshsoni9263 (Sorry for my bad English. I have to use Google Translate) The triangles are similar because all the opposite angles of the triangles are equal. One angle is 90 degrees and the other angle, where the line connecting the ends of the pillars intersects the ground plane is equal as well. When there are 2 congruent angles, the third angle is also equal. So the shape of the triangles is similar. I hope you understod what i ment

[@gregorymagery8637]

@@jigneshsoni9263 (Sorry for my bad English. I have to use Google Translate) The triangles are similar because all the opposite angles of the triangles are equal. One angle is 90 degrees and the other angle, where the line connecting the ends of the pillars intersects the ground plane is equal as well. When there are 2 congruent angles, the third angle is also equal. So the shape of the triangles is similar. I hope you understod what i ment.

[@gregorymagery8637]

@jigneshsoni9263 - This is not a proof but you can also think of angles like this: Light takes the SHORTEST PATH from one point to another Imagine that the ground plane is a mirror and you reflect a thin laser beam from the tip of one pole so that it hits the tip of the other pole. Light reflects from mirrors according to a certain rule. This rule is called the law of reflection: the angle of incidence is always equal to the angle of reflection. => 90 degrees + angle of incidence = 90 degrees + angle of reflection => So the third angle in triangles is also equal. => So the shape of the triangles is similar.

[@mihaip56]

In Euclidean geometry, it is the postulate that states the minimum distance between two points which is the straight line. If you build the projection of the 28ft line in the lower plane you will obtain a right triangle with legs 30ft and 40ft, you will discover the hypotenuse having 50ft (Pythagorean numbers 3, 4, 5 multiplied by 10), Thus you have discovered the minimum length of the string, And if you are curious how far the point is from the origin you will have to remember the phenomenal mathematician Thales of Miletus. It remains only to wonder how these ancients managed without Desmos and without Microsoft Math Solver!?

[@thebluegarden2060]

My problem is the bottom leg is 30 feet across. Where did you get 10 and 20? It could be 9 and 21. It could be an infinite amount of length that total 30 feet. From the drawing, it looks like it’s centerline which will make it 15 and 15 making each angle 45°. Please answer me this. Where did you get 10 and 20 for the bottom leg?

[@Anu55203]

Thats why he took x and 30-x.

[@larryevans2806]

He used 10 and 20 to initially solve a hypothetical problem, i.e. "let's suppose the distances are 10 and 20; what would you do to solve the problem". That helps to formulate a method of solution.

[@debunkosaurus8228]

Another way to solve this would be, after taking the derivative, recognise that the square roots are just Y and Z. This will give you similar triangles which becomes easy to solve.

[@HOME-GH]

without using numbers for the poles and representing the heights as H1 and H2 separated by a distance D, and x being the distance from H1 to the meeting point, and D-x from H2. Set up an equation for the total Length L: dL/dx= x/(sqr(x.x--H1.H1)) minus D-x/(sqrt((D-x).(D-x)--H2.H2)), equating this to be equal to zero results to tell you that the two angles must fit in the case of similar triangles making is similar to use ratio and proportion

[@tomtke7351]

T H A N K Y O U -- for others... review the "chain rule" before doing this problem. I'd completely forgotten it.

[@Jedermeister]

basic optimisation problem. fun one. My further mathematics students will like this

[@peterweusten4251]

Can do by reflection of 12 ft pole in ground level and join top of posts. Good to check answer

[@kurtsopa]

The length of wire is 50. Please correct me if I’m wrong.

[@ChristopherGutierrez]

You don't need calculus for this. In fact, you can do it with basic geometry. A wire from the top of the 12ft pole to the top of a 28ft pole is the same length of wire as from a 12ft pole to the "top" of a 28ft pole that is upside down (ie underground). The difference is now you can consturct a single triangle with one side at 40 (12+28) and the other at 30. This is a 3,4,5 triangle, so the wire is 50ft long. Because the angle this the smaller triangle is 3,4,5 or 3 * (3,4,5). So 3*4 (the pole) 3*3 (the length from the base of the pole to the wire reaching the ground. 3*5 length of the wire from the top of the pole to the ground.

[@chrisboonzaier7813]

Yes you can, but the whole purpose of this exercise is to practice and learn calculus.

[@ChristopherGutierrez]

@chrisboonzaier7813 Why create problems you can do in your head without calculus, when there are so many interesting problems where calculus makes a complicated problem easy. Seems counterproductive.

[@FredCarney]

I have a problem that I'm not sure how to set up. I'm using a 5 gal bucket as a rain gage. Buckets are tapered so that the diameter is smaller at the bottom than the top. One inch of rain that enters the top is more than one inch in the bottom. Each succeeding amount of rain measured gets closer to what it was entering. The last rain to fill the bucket will be equal to what it was entering. I don't need the volume of the bucket. That is a simple frustrum of a cone measurement. I want to dip the bucket and calculate directly how much rain entered the bucket. How would you solve that? Thanks

[@TristanBBBBB]

Has anyone ever made it through one of your videos? With best intentions, I suggest less repetition. This could have been just as illuminating in 5 minutes.

[@zodeseeker]

You are an expert in math but you might want to go down the hall and sit in on an English class. At 2:55 you said " How important algebra and geometry is" . In this statement the is should be "are" as the noun is plural not singular. You don't want your students disregarding your math instruction because you communicate in an improper manner.

[@barryclements62]

Some of your solution was fuzzy owing to how many years ago it was taught. Is the length of wire then 50 ft?

[@PaulZimmerman-t4k]

This guy couldn't teach a dog to pee on a fire hydrant

[@Hobby.Turner]

the angle where the rope hangs is the same both sides. tan a = x÷12, tan a = (30-×)÷28; x÷12=(30-x)÷28 28x=12(30-x) 7x=3(30-×) 7x=90-3x 10x=90 ×=9

[@dhy5342]

"the angle where the rope hangs is the same both sides." - how do you know this?

[@kel-in5gi]

I don't understand @ 32:00 why you don't distribute the negative sign across the denominator.

[@donnag6902]

A negative distributed thru the numerator is like multiplying by (-1/1). If you distribute a negative thru the denominator, also, you are multiplying by (-1/-1) which is a positive one.

[@noferblatz]

No problem on the length. This is a practical application for calculus. I would never have known that calculus was a method of solving the problem.

[@lengthmuldoon]

After 50 yrs I now understand a practical use for calculus - nice work

[@JohnBerry-q1h]

For decades, high school Physics classes have been presenting Calculus equations to students who haven’t taken Calculus. Why apologize now? To do so would be to mess with a time-honored teaching tradition.

[@glogenuniversal6047]

Say I want the maximum length of wire?

[@ohary1]

What if you just put a weight between both points and measured down to the lowest height?

[@chamberizer]

It's like he explained how to use calculus, but solved as a quadratic. It might be a combination of both? I think - I may have to look at again. My wife wants to talk to me at the same time, so it is hard to concentrate.

[@carloheinz6465]

Distance on left....12/40×28=9 or distance on right...... 28/40×12=21. (Test: 21+9 =30.). No need for such an elaborate approach 😬

[@LongDogRacing]

there is every need when trying to teach CALCULUS....

[@malekal-omary6269]

they have the same slope 12/9=28/21or 12/x=28/(30-x)

[@ralphhardie7492]

😂😂😂 Catcher in the rye teacher

[@laurencepeterson]

much too compliicated to explain basic calculus: reminds me of an old joke: "i ask you what time it is, and you tell me how the watch is made>'

[@joem9380]

So how much wire do you need 😮

[@ARrocks12]

You made a mistake at 25:00. It's (2x-60). You forgot the brackets

[@JasonKing-m6m]

Is it possible to learn Calculus on it's own without first having to learn maths?...

[@ahzzz-realm]

The small triangle, shortest right angle making 12 the b side. x is 12/4=3, making the 'a' side 3x3=9. c=15. The large triangle is a x b, is 21 x 28 = c of 35. 15+35=50 ft cable.

[@drdotter]

12*(30-X)=28*X; X=9

[@peterreali3950]

Lots of ways to make arithmetic mistakes but he didn't give how much wire is needed and its 50 ft at 9ft from the first tower.

[@erickchan2871]

How did it appear 10 and 20 it should be 30 - X

[@Intense_Cloud]

He did it for pure demonstration as to prove for the use of calculus further in the video.

[@kenhaley4]

For someone just learning calculus, I think this problem is way too daunting. The instructor even had to skip a lot of the algebra details to get to the final result. Plus, to find the derivative, we had to use the power rule (twice) and the chain rule (twice), in addition to understanding negative fractional exponents. Even if well understood, there are many opportunities to make mistakes, which could be pretty discouraging for a newbie calculus student. I think any student with sufficient skill and background to understand this video, would also realize the problem can easily be solved without calculus as other commenters here have pointed out. Most beginning calculus textbooks have better problems to illustrate the point of this video without the complexities of this example problem.

[@fred8780]

too complicated. 12/28=x/30-x; x=9; 21; pythagorean and done. trig is enough.

[@StevenStyczinski-sy8cj]

So the slopes are 4/3. 12 high by 9 long: 28 high by 21 long: 12/9=4/3: 28/21=4/3! O K , next set of poles.

[@alocin110]

You did not use the pythagon theorem to calculate the length of wire to prove how long it is compare to the first two points. so this video is not complete and it does not prove your point.

[@DancingPony1966-kp1zr]

Beautiful and accurate, of course. But you lost me way back there with all tose steps and big numbers. The trigonometric approach above seems quicker and more intuitive.

[@balimbula]

Yup. Calculus is a discipline. Today AI could figure this problem in seconds.

[@johnupdates2472]

Or just use the easy calculation x/12 =30-x/28

[@jayantaboral6446]

Sir dw/dx = 0 is not the only condition to find local minima. d2w/dx2 to be also checked for > 0 condition for minima. The solution you discarded is also another solution which gives local maxima (i.e value of x for maximum length of the wire) as d2w/dx2 would be < 0 for that value of x. Both maxima and minima satisfy the condition dw/dx = 0 Also you did not calculate the length of wire (=y+z), it simply finished at calculating x! 😂😂

[@russelllomando8460]

Wow...

[@gbenother8755]

Surely this guy must be the worst Math presenter on KZbin. His repeated mispronunciation of PythagoreaNNNNN just drives me nuts!

[@chriswright9819]

its pythagorus's

[@trendwin2024]

There’s a better way to solve this problem!

[@dnwiebe]

Py-thag-or-e-an, not Py-thag-o-rim.

[@MrMackxl65]

This not a Calculus problem; it's a Similar Triangles problem.😂😂 Just minimize the the ratio of the hypothenuses of the small and large triangles.

[@chriswright9819]

it.s