Calculating the Work Required to Drain a Tank

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) / patrickjmt !! Calculating the Work Required to Drain a Tank - Using Calculus! One complete example is shown along with a general procedure to follow!
For more free math videos, check out PatrickJMT.com

[@patrickjmt]

i met my girlfriend, and now wife in calculus 2. it was the only reason i stayed in the class! she now has her phd math, and it is totally hot. : )

[@asemnafiz]

Patrick has this amazing ability of simplifying complex maths problems into easy words. Not everyone can do that. Many people know how to solve a math problem but cannot express that into words properly.

[@danielkim4502]

@manhong liu, except the part about solving as well

[@spitforge8963]

I love him

[@loaf8506]

10 years later this video is still helping people!! i missed this day in class and my textbook was a bit convoluted but this cleared it up. thanks man!!!!!!

[@elizabethmendoza1990]

I completely agree!

[@patrickjmt]

ops, you are correct : ) that just reflects the weakness in my geometry... it is not an equilateral triangle!!! thanks for pointing that out... no one else ever has!

[@angeljohn1250]

PatrickJMT and khan academy are one of the most useful sources of education in my life

[@patrickjmt]

yes, the volume would have an impact on the weigh (that is, the force) involved

[@abigaillindner]

Thank you for this video, Patrick! I'm reviewing calculus from previous years in preparation for my first year of college math (Calculus III), and this procedure in the book wasn't clear to me no matter how many times I read the section. Your tutorial has helped me gain a better understanding. All the best!

[@torment0fsin]

youre awesome. i have arguably the best math professor for calc 2 at my university, and youve explained better in 10 minutes what i failed to really grasp in 50 mins lecture, plus office hours, plus group study sessions. i am so grateful to have access to these videos

[@kurndoge3226]

Who else has an exam over this tomorrow?

[@mushovers3006]

not I

[@sal20Jets]

Me!

[@jordanadler6853]

I have an exam in a few hours lol

[@christianjmyhre]

Today actually lol

[@davidbourgholtzer788]

In one hour

[@laurencrom]

Why can't James Stewart be as straightforward at PatrickJMT?? Thank you for, once again, being my go-to source for simple explanations!

[@tahmidbhuiyan5591]

12 years later and this still helps students today! This is the beauty of teaching

[@imneveru]

How is it an equilateral triangle if the height is 3? Shouldn't the side be 3? So the height would actually be 1.5sqrt(3).

[@carlosfernandez1989]

I went into calculus two with no knowledge of calculus one and lacking serverel years of practice in any math. I for the longest time had the lowest grade (20%) in my calculus 2 class. But after finding these videos and drilling the problems, i was able to raise my grade to a 91%!! Your videos have been amazing, great work!

[@alexisaquino2777]

How did the math help? 👀

[@pieguy49]

can also do width=x and instead of displacement x+2 use distance 5-x. For that x=3 at the top of the tank and x=0 at the bottom.

[@scien-tasticvideos2398]

This channel is a life saver

[@techhungry1]

@patrickJMT I started watching your videos because I had a test coming up. Before I watched these videos I knew nothing, I got the test back to day and 96/100. Thanks again!

[@jonnathannickolai7827]

damn patrick you make my college tutoring SOOO EASY to explain...

[@SidneyBloom]

The key here is that the force needed to pump the liquid does NOT depend on the distance the liquid travels, as the weight of a certain quantity of liquid is constant. But it DOES depend on the height of the liquid because it is related to the size of the slice of liquid we are considering. The limits of integration are the max and min values of the height of the liquid, and not the displacement of the liquid. I think it's more appropriate to use the height of the liquid as the variable, and not the distance from the top of the tank, even if you are going to get the same result. I think that's why people are confused.

[@patrickjmt]

@Hewsonx the weight of water already accounts for that

[@patrickjmt]

nope, since 'x' represents the distance from the top of the tank to the water level, it should be [0,3].

[@eliezerr.johnson9644]

Thank you so much for creating this video. Work is probably the most challenging part of Calc ll, and the instructions in the book skip so many steps... Again thank you, you make it seem more comprehensible.

[@EmBennet99]

We're using this textbook and these videos are a lifesaver. You're an amazing teacher.

[@OlceOktavia]

the displacement, shouldn't it be (5-x)? or maybe we'll get the same result with (2+x)?

[@WhatTheWeb]

no it can't be 5-x

[@coolpig96]

I'm still quite confused with this topic but I'll work it out myself :) Thanks for the video. Although it has been up for 7 years. God bless you!

[@patrickjmt]

yes, u are very correct - a few other people have already commented about how bad my geometry is - which i readily admit

[@patrickjmt]

@geovaniluna no, but i should make one. i probably will... eventually!

[@MrEyee2]

Tired of scratching KZbin for math help?...Hmmmm no, not when Patrick is there.

[@coltonleeware95]

That is not an equilateral triangle.

[@Axeleration]

Thanks a bunch for the video, I've been trying to understand the similar problem in Stewart's Calc book. I really like your approach with the f and displacement, it makes the integral setup a lot clearer and simpler.

[@Shredderbeam]

A brilliant video! Very easy to understand, well-explained - resolved all questions I have on this part of calculus. 10/10

[@insanemxrider1]

extremely thorough!!!!! this is what makes a great way to understand calculus thanks i have a test today and this is helping.

[@vincentarrage]

after watching about 500000000 of your videos, I just realized you're left handed

[@mango416]

I believe that it is 0 to 3 because it is the water affected that is shifting. The height of the spout is not considered because there is no water there. We account for pushing our water to the spount with (x+2) as x represents the distance we travel down it is like a generic distance

[@patrickjmt]

@sportmanic50 yes, you are correct. i thought i added annotations about it but maybe not

[@patrickjmt]

thanks!!! glad i could help

[@lolan5043]

Thank you, Patrick!!! You make calc look soon easy!!!! Everything just makes sense the way you explain it. thank you!

[@davidontiveroz8295]

No one will probably ever read this, but i think there's a mistake... maybe. i think the displacement of the water = (5-x); not (x+2) help anybody??

[@jdlonger]

I believe Patrick is correct, it should be (2 + x). The water has a minimum displacement of 2 ft, and maximum of 5 feet. Therefore (2 + x), Where "x" is greater than or equal to 0 and less then or equal to 3.

[@AllStarTeamFlight]

thats what i was thinking other examples of this problem do the displacement 5-x... whats going on here?

[@patrickjmt]

(x+2) is correct

[@TurtleSensor]

patrickJMT In the book they used (5-x) but they also did it slightly different. I did the same calculations that patrickjmt did in the video and got the same answer.

[@tofo7151]

***** depends where you put your x

[@stepheng.6145]

The displacement is (5-X). He didn't account for the top of the spout to the bottom of the tank. The real answer is 1058.4x10^3 or approx. 1.06x10^6 Joules.

[@xXxDragonbossxXx]

just jumped on your video by mistake .... it looked strange and hard in beginning , but you did excellent job in explaining algebra part ... thank you i understood something now .. i need a lot of calculus video like that will help my carrier a lot

[@1matth3w1]

Respond to this video... your vids are really effective, ive been watching you for a while when you had 3000 subscribers.. you have become popular.. and arguably the best out here.. god bless

[@qwertyguy76]

Thank you soo much, let me say that PatrickJMT is the best youtube calc teacher out there, and yes this is from a James Stewart book lol.

[@cestlavie9933]

Where did he get 62.5 from?

[@LetsGetItKhan]

density of water, but you probably don't care anymore lol

[@visionary6498]

I care bro thanks

[@danielpatino4010]

@@LetsGetItKhan The real MVP

[@michaelthrasher5900]

cestlavie9933 that is the density of water

[@patrickjmt]

@1matth3w1 thanks for the support : ) yes, things have grown quite quickly! it blows my mind when i think about it. and you know, people are so gracious and thankful; it makes me happy to be able to help in this way.

[@wssz112]

patrick u r so amazing ... all of my math professor love to trick us instead of "teaching" us!!!

[@heraallen9707]

you are MUCH clearer than my professor and I have a final tomorrow, so thank you thank you thank you!!!!

[@RuvieEto]

@Ackeem91 JMT is right. He used the 62.5 to convert it to a force. He is working in imperial units (ft, slugs etc), so instead of 9.81*1000 he used 62.5 , like he mentioned at the beginning.

[@rocket6923]

I don't agree that the displacement is 2 + x because the water not in the very front of a given horizontal slice has to first travel forward to get to the spout. If there were spouts completely covering all edges of the tank, then I would agree that the displacement is 2 + x. But because this is not the case, the water in the very back, for example, has to travel 10 units forward first.

[@TimpBizkit]

@johnallenasdf technically it's not an equilateral triangle but as he wrote the question using a height of 3 not a side length of 3, and stuck with that through the calculations, (even though he did have a slip of the tongue) it didn't affect the answer. By the way, the height of an equilateral triangle of side length 3 is 1.5*sqrt(3) Yes I know you've probably found the answer by now.. 2 yrs

[@oscarsandoval2324]

kwok is right...besides there is more than one approach to this problem, if your using similar triangles you can either account for displacement immediately when finding the width or account for displacement(distance to move slice) later...the latter is what I prefer because it allows me to conceptualize better although sometimes results in a messy integrand..whichever results in easier integration eventually takes the cake..problem # 24 is a perfect example of that scenario.. not sure if you got the right answer dude but I'd be lying if I told you your video wasn't helpful. Thanks

[@paintballer7509]

setting up the volume, the way most books do, would get you to the integral of 62.5 * 10*x*(5-x)dx from 0 to 3 which turns out to be equivalent what was done in this video. It all depends on how you label the diagram

[@techhungry1]

your videos are the best ones out there

[@patrickjmt]

@FWDorRE yup

[@kool45520]

How can the height possibly be 3 ft when it's an equilateral triangle?

[@enid1015]

From where it comes that if it is in feet you have to multiply by 62.5 or if it is in meter you have to multiply by 9,800?

[@nsbchildrenofthecorn]

whether you have your function being equal to 625(5x-x^2) or 625(6+x-x^2) all depends on the x axis you superimpose, you get the same answer :)

[@ian559fresno]

It seems that every time I need help with Calculus Patrick has a video on it :)

[@thok1599]

Thank you very much! Better than my teacher.

[@Juxtaroberto]

In a problem involving a cylindrical tank, yes, but since this is a triangle. and the width is changing at every point along the height, then no, it's not the same.

[@patrickjmt]

they can be tricky problems, so do not get discouraged

[@resistnzisfutl]

Thank you so much for all your vids, they are one of my best resources! So about this video, shouldn't the displacement be 5-x rather than x+2? So, if the slice was at 1 foot at a given moment, at x+2, you'd get a displacement 3 feet from the spout, which is wrong, but with 5-x, the diplacement from 1 foot is 4 feet, which is correct. Or am I missing something here? Thanks!

[@meheeners]

I have the single most aggravating calc professor in the universe, because of your videos I have a B! You have saved my GPA.

[@patrickjmt]

@techhungry1 congrats! that is a good score!!

[@signinname41]

Also, why isn't acceleration due to gravity included? I thought force equals density x volume x g.

[@mariuszwiesiolek5777]

This is amazing, especially the explanation of similar triangle theorem!

[@maggiehernandez3126]

When all else fails, here comes PatrickJMT to the rescue! Thanks!!!

[@avpescatore]

This man is a legend.

[@devinziehmer170]

see i thought it would've been (5-x) but in technicality he's just accounting for distance x + the distance of the spout for how much force is going to be required to drain the water from the water level up, because, essentially that distance shouldn't be changing as far as where the water is when it's being drained that distance. You can't say (5-x) because x isn't the height of the water, it's the distance between water level and the top. So to say (5-x) would be to say the entire distance of travel minus the distance from water level to top. This is why you incorporate (x+2) to determine your distance so you can plug in force x distance to find your work.

[@soccerforlife1111]

Great video! You have a wonderful gift and I'm glad that you share it with the world. One complaint. you say that the tank is supposed to contain a face with an equilateral triangle. If one side is 3, then the other sides are also of length 3, making your height (3√3)/2, not 3. So either your height is wrong, or you don't have an equilateral triangle. I would assume that it's not an equilateral triangle. It doesn't change any of the math in the problem, but I thought I would point that out.

[@nikan4now]

Alternatively you could calculate the total weight of the water in the tank and place it at the center of gravity. The work required would be the weight times the distance from the center of gravity to the tip of the pipe.

[@xLouDNoiSeSx]

Couldn't understand my Japanese Calc 1 teacher, can't understand my Korean Calc 2 teacher, can understand patrickJMT.

[@yasmeenkarachiwala9612]

Limits of integration should include the height of the spout lower bound 2 upper bound 5?

[@charliechickie]

this was very helpful, except shouldn't you use y instead of x since the water is moving vertically? My Calculus teacher has us use dy when we integrate for pumping

[@ahmedalshangiti325]

All I can say is really Thank you!!!!!!!!!!!

[@Daniavila40]

thanks a lot! this is really helping me to prepare for my final exam

[@TLL228]

Hey man thank you so much for this video, I'm in calc 2 in college and it helped me extremely!!

[@Nickuncle]

Because he chose x as his variable. It's not on a plane so it doesn't matter what variable you choose. You could make it a dg integral if you feel inclined. Feel free to correct me if I'm wrong.

[@patrickjmt]

it is roughly the weight of a cubic foot of water

[@Daniavila40]

Thanks a lot!! This is really helping me to prepare for my final exam😃

[@mlapinski7]

i think i'm gonna ace calc 2 next semester. thanks patrick! and grats on the hot math wife ;)

[@himomhey3437]

did you aced your calc 2 in your semester?

[@charliechickie]

also, to answer nelmsters-filling and draining is the same amount of work

[@TheChondriac]

When doing the similar triangles, why is the height of the small one (3-x)? What if we started x from the bottom? How would it be solved then? Would the height of the smaller triangle just be x?

[@signinname41]

Do you ever have to account for the tank being a triangle instead of a rectangle? I don't see anywhere in the integral that makes the rectangles get smaller towards the top to form a triangle. It seems like these numbers are for the work on a stretched out cube rather than a triangle.

[@emilyunderscoremarie]

Thanks bro....these videos have been saving my life when both my professor AND t.a. are foreign

[@builderk]

It can't be an equilateral triangle, since the height is the same as the width. If it were equilateral, the height would be (3*sqrt(3))/2, or ~2.598.

[@feezy243]

Can you explain why the limits of integration isn’t 0 to 5 (water has to go through spout)?

[@patrickjmt]

if you think about what ' work ' is, you should be able to answer it without doing any calculations though

[@foxpizzaguy]

@Ackeem91 you only use 9.8 for gravity in metric units

[@seth.heerschap]

I love the internet... theres no need to buy $200 textbooks, when you can just watch these awesome videos

[@tahoeisgreat]

the height of the triangle isnt 3 if its an equilateral triangle, and one side is 3.

[@Jfrancis221]

couldn't we use 62.5 x integral[(5-x)(10x)dx].... (10xdx is the length) (5-x) is the full distance from the spout to the bottom of the tank. (3-x) is just the triangular part of the tank... the "slice" of water still changes on the way up through the spout in order to pump out the water correct? we can still integrate from 0 to 3 because of that being the "actual volume" of the tank ..then multiply it by 62.5( which is the weight of the water .. which is also the force since water is a force).

[@iandenaro]

@4thKyuubi its a given. water weighs 62.5 lb/ft^2

[@patrickjmt]

@rr5001 yup

[@hunterh8054]

honestly better than my professors

[@sekhelmet4]

On your website under the topic of draining the tank the link takes you to a surface area video just letting you know

[@patrickjmt]

yep, that is the idea.

[@tjsumpthun]

It is 0 to 3 because the height of the triangle is 3. So do get all the water out of the container, you must go 0 to 3.

[@robwat5370]

Where does gravity constant come in, doesnt Force = M x G and so by my way of thinking your figure for Fslice should me labeled Mass and multiplited by gravity before becoming the value for Force. Forgive me if im wrong im just starting out with this stuff hence why i ask the question

[@squashiballs]

well.. if you take downwards as positive... shouldn't the displacement be -(2+x) considering that the water moves up?

[@eunkuk91]

btw u put up a wrong video for this section on this section in ur site... and could u explain when there is no 'tube/hose' on the top...? and always appreciate your work.

[@BrandonByerly-Sam9501]

Why do you have to find a general volume for a slice in the tank? Is it possible to solve the problem while using the total volume of the object defined? Isn't it more practical to use numbers that are already defined to create the object? I'm confused about that aspect of the math and if anyone could help me, that would be greatly appreciated.