Related rates: water pouring into a cone | AP Calculus AB | Khan Academy

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As you pour water into a cone, how does the rate of change of the depth of the water relate to the rate of change in volume. Created by Sal Khan.
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AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.
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[@adam-uk9wp]

hardest part is just setting up this shit. finding the derivative is easy but getting the equation to differentiate is such a btich

[@winterwraith8339]

Adam Resh just remeber the volume equation for shapes. practice those on your free time. do it like a hour a day for a week. once it's melted in your brain remeber you take the volume of the substance and BANG!! your golden. you can find all the pieces you need in the diagram n do the easy part. practice doing a few more problems like this and it will start getting easy ;]

[@tomatosteve3444]

I too will use calculus to find how much water I have in my 4 cm tall conical shaped cup at 2 cm of water when water is running at 1 cm^3/sec

[@johngddr5288]

I do that all the time!

[@Brandon-ms2uc]

Or when you’re pouring sand into conical piles.

[@eliotwilliams4480]

I like the sound of that! Maybe I should practice doing it more often.

[@abdulazizob]

the drawing is a masterpiece.

[@ericcartman1168]

My first thought

[@eliotwilliams4480]

quite soothing.

[@Xrdnk]

I can understand when other people do it but when i do it i get wrecked

[@orbital16]

Hahaha me too ..

[@fanimeproductionst.v.3735]

😁😁😁

[@footballeditzs4957]

Just more practice in your free time not exam conditions

[@megan2118]

@@Houndrag This is going in my notes, thank you sir

[@ImElytra]

@@Houndrag u r a pogger sir

[@ozzman1997]

I was waiting for him to correct the pi / 2 thing, lol. Seems like an intentional mistake my calc teacher would make to see if the class was paying attention.

[@joolee_ette]

I WAS SO CONFUSED

[@sienan]

lmaoo my teacher does the same thing..

[@kinza7549]

holy shit drawing skills on point tho

[@shashankmurthy5668]

+Kinza R Right, dude is good at everything

[@ericm1839]

btw, this has a 50% chance of showing up on your AP calc exam (both AB and BC)

[@kittyrules]

Thanks bro

[@gamerstheater1187]

I've seen it five times in our homework

[@forevereveryours]

Who knew Sal was an artist too, oh dang.

[@dichiethiel7201]

Hannah he is everything

[@charlesluo3114]

Cool vid, for anyone confused, when you take the derivative of the rearranged volume formula, you can just tag on the dh/dt next to the new derivative of height you just found instead of elaborating with the h(t) part.

[@lasseokaty796]

Life saver thx made it way more confusing than it should be

[@mozacchi]

drawing tutorial link?

[@RocksYourFace44]

That diagram is very khanical.

[@safakguney6912]

this is one of the best comments on youtube that I have ever read :D

[@robbyburns5822]

needs more likes

[@mrayushtiwari]

Awesome dude😎😂😂

[@sheepycultist]

god Im so fucked

[@LR-ce9vp]

lmao, bro, honestly do a few easier ones to build confidences.

[@ellieb8627]

Louise Hsu quiz tomorrow.... rip AP calc grade

[@lusamini]

Louise Hsu my thoughts exactly😂 I'm gonna fail this class! How'd you guys do?

[@huntersmith6821]

NO

[@AkidsukiLeader]

I give up!! If Khan Academy can't help me to understand it then, I. Give. Up.

[@hawkeye2958]

I find the less you try to think about it theoretically, the easier it is

[@aretwodeetwo1260]

same thing here, it seems like all you do is toss a bunch of equations into a bowl and stir them all up. Makes no sense whatsoever.

[@zanytooth]

It is supposed to be 1=pi/12*3*2squaredh/dt

[@ellieb8627]

"Let me make it clear" yes because writing it in some weird way makes everything so much clearer hahaha

[@lmfaowtfinghell]

Ugh... I understand it when he does it but I can't figure it out by myself :'(

[@benleighton1709]

+lmfaowtfinghell My life

[@firstcomesrock8593]

lmfaowtfinghell for real calculus is ridiculous

[@japagordo6564]

you just need to practice more

[@joy397]

i watched this whole video and absorbed nothing

[@nibs991]

Lmaoooo i took away a little bit but more power to ya

[@saiyanbro]

He’s making this much more complicated than it needs to be. Other videos simplified it so you can actually learn

[@Wolfie020]

Thanks so much for this video! Never quite got this concept, it will be very useful on my upcoming Calculus AP test.

[@uchihamadara6024]

Got an exam tomorrow and this is the one thing I didnt understand. Thanks khan academy, for saving my ass year after year

[@Enigma758]

I think the problem is even simpler if you use the chain rule this way: dV/dt = dV/dh * dh/dt. Once you have the formula for V(h), take the derivative and replace dV/dh with that. You're given dV/dt so plug that in too, then all you have to do is then solve for dh/dt.

[@hunterdixon5845]

I love this, it helped me understand how to take related rates. I couldn't have done it without ya ;D

[@LeeJordanMusic]

Why were you able to take the constants out at 5:36?

[@cameroncabell9871]

Because you're multiplying the constants by the derivative, not taking the derivative of the constants

[@emperordarthjarjarsnoke7596]

Constant multiple rule.

[@arcadianproxy3730]

anyone else internally screaming when he wrote pi/2?

[@camo736]

pi/12 right?

[@Appl3Muncher]

Thanks khan academy has helped so much.

[@philisdead4]

WHAT THE FUCK?

[@Punjabispitta]

hahahaha I literally just said too, "I wonder how many wtf's I'm going to see in the comments" lmfaooo

[@JagjitBrawler]

+Harnoor Singh sat sri akal

[@k_______p]

Thank you very much!

[@yooncho2112]

Thank you so much!!! I finally got the answer for a similar problem I struggled for hours!!

[@kr3wdog06]

He wrote pi/2 instead of pi/12......

[@arcanehack]

you would think the amount of times he repeats stuff he would have wrote it lol

[@Christianturner7]

He corrects it at 10:28 relax.

[@zappist751]

GREAT EXPLANATION SIR KHAN

[@LadyHelenaRavenclaw]

that diagram is a work of art

[@behnamasid]

What an amazing answer.

[@keotshepilemandona9882]

the guy talks for 11: 31 minutes and change my whole life...and not only me and for millions and billions of people out there.....

[@lil_yerrr]

Keotshepile Mandona this video on related rates really changed your WHOLE life?

[@gelatinocyte6270]

So far (at the time I reply this), there are only at least 600k views in this video.

[@jimcar53]

As usual - Brilliant

[@famine784]

Just the question i needed help with THANK YOU!

[@johnnolen8338]

Hint: For a cone dV/dh = π r^2, in every case. For this case r/h = 2/4 => r/2 = 2/4. Therefore r = 2×2÷4 = 1. dV/dt = dV/dh·dh/dt = (π·1^2)dh/dt. Hence dh/dt = dV/dt ÷ π = 1 ÷ π centimetres per second. ◼

[@TheIkarus93]

sal you should make some modern physics vidoes; relativity, atom models, intro to quantum physics and stuff PLEASE

[@creatureunderthebed7083]

Not sure about you guys but I was DEFINITELY getting a tingling feeling about the chain rule there...😃😯

[@hakalakalaka0.963]

let my understanding of related rates=r. Find dr/dt. dr/dt=0

[@milesweiss858]

+Hakalakalaka1 so, your understanding of related rates is constant! better hope it's a high number.

[@NightbladeNotty]

so can I just move in with u and study until I get my engineering degree? ur a lot more understandable than my professors haha

[@kg7111]

how'd it go

[@NightbladeNotty]

@@kg7111 graduated finally before the pandemic lol so I got to walk at least 🤣

[@FlamingAlchemist7]

Did this in Calculus not too long ago

[@kyuujuuwang]

that was magical

[@ismaeeldasti2773]

Pedro, no because when u differentiate that u get dr/DT and we can't have 2 unknowns in one equation

[@sarikasabnis2369]

Sal, you can teach me anything and your drawings are pure masterpieces.

[@clueless.2509]

thx

[@djancak]

great explanation, thanks

[@raymond-reviews]

Could you guys do related rates for troughs

[@Frances3654]

very interesting

[@mtdeezy]

This could be done really easily, though. The rate of change of height is the rate of flow divided by the area at the height. In this case, the radius is 1/2 the height, and the area is pi times the radius squared. So the area as a function of height is 1/4pi*h^2. At 2cm high, the area then is exactly pi. And the rate of change of height is flow/area =(1cm^3/s)/picm^2 = 1/pi cm/s.

[@Nike7548]

Thanks bro lots of help :)

[@RTRVII]

No he is only interested in the momentarily change of h so what he does is correct. To get the complete function h(t) he must indeed solve a differential equation with a satisfactory boundary condition.

[@he0k0n41095]

should fix it to pi over 12 man......

[@vgoj]

Mistake at 9:35 . The h is a function of time. You can´t assume it's equal to 2cm. It's only equal to 2cm at the instant t=0s. To solve this problem one should use a simple ordinary differential equation (ODE) to find the function h(t) and V(t).

[@luiscecenas6681]

If learning math Khan explains it easier and better then all the teachers even college teachers😅

[@Lauderdalesfinest954754]

Fluid Dymamics..! Cool..

[@lancelovecraft5913]

h=d the diameter 4=4 and 2=2. r is d/2 or h/2

[@RobbyBoy167]

Hey Khan, i think it would be easier if you'd just go with the idea of implicit differentiation.

[@MrAlbashiri]

thank you very much for the video. helped a lot :)

[@Crob56]

Something with respect to something and the derivative of something.

[@thoughtprocess-]

I love you videos. thank you

[@9starbrite3]

He fixed it after a bit.

[@RogerKlauser]

At 8:27 the you wrote pi/2 but it should be pi/12.

[@HenggaoCai]

Sal, I challenge you to do this problem. I bet you can't do it. A) "Water runs into a spherical bowl of radius 10 ft at a rate of 25 pi ft^3/min. How fast is the water level rising when the water is 6 feet deep?" B) "Find the depth of the water when the bowl is filled to 2/3 capacity"

[@yashuppot3214]

How would u solve for the deriviative of h with respect to t in general

[@trevordavis-dressner9913]

This is a function of time, that has respect to time, within the derivative of t, and the chain rule with respect to time

[@TheMONOPOLY123]

that drawing O.o

[@ShaMan3D]

How you convert linear motion into circular ???

[@HenggaoCai]

These problems are so easy. You should see the problems my teacher gives us. They are 3 times harder than the AP exam

[@Sp0okeh]

i love the drawing of the cone and faucet. unfortunately im still trying to figure out the problem...

[@FireVortex720]

thnx man

[@joshuacoleman714]

he corrects himself if you watch all the way through.

[@MrKushinator420]

No matter what i do, I can't understand all the random problems they throw at me. Neither one helps with any of the others. Think i'll just stop trying this bullshit.

[@emlmm88]

Have you had the baseball one yet? Or the gradually deflating balloon? Fuck calculus.

[@MrKushinator420]

I managed to get a C in calc 1 and a B in calc 2. After that, I took linear algebra and got a B+ (way easier). I don't think I'll ever be working with calculus though.

[@kumis91]

where you get v=(πh^3)/12 if you get the derivative of that then you have dv/dh and if you get the inverse of that you get dh/dv and then you can times dh/dv*dv/dt and you get dh/dt and then just sub in 2 for h, i find that less complicated, even though it'll look way more complex in this notation

[@hardikguptaiit-k5028]

Nice video

[@TheFaisal00000]

man give me the the answers. How can I know that if I'm correct or wrong? Answers will help me a bit too.

[@shanebauer8091]

Why are you able to take the 3.14/12 out, if I may ask.

[@jakeengel9377]

I think it would be better if you were to explain the mentality behind the concepts you are using in the actual problem, especially because every example is different. I think most people understand things best when they are given a reason as to why they’re doing something, such as just finding the derivative of both sides of the basic area of a cone in respect to time, then plugging in the values that are given; Always dumb things down as much as possible, it will make more sense later

[@17puttab]

You meant pi over 12 at 10:13

[@mingyan1966]

he corrects the mistake later at 10:23

[@SnipeSniperNEW]

when you fill a cone with water, water height and radius increase if its rate of change of its volume is constant. the rate of change of the speed of increasing of the radius and height is decreasing (logically due to the cone shape) so how can we calculate this acceleration of in this case deceleration.

[@adolfocarrillo248]

Here an interesting question is How much is the dH/dT changing each moment. its much more useful.

[@yashuppot3214]

So the second deriviative of the height with respect to time?

[@Nikifuj908]

Is that an exact measurement? Does it corroborate with other results? What is your sample size? Are there any outliers that could skew the data? How did you test problem difficulty- with human subjects or using a metric? Also, did you know that the purpose of Sal's channel is to teach, NOT to do the world's hardest problems on a dare?

[@nonchalantd]

I hated this problem when I was struggling to learn how to do it.

[@zag5434]

I have a question asking for the exact same thing with different numbers and I somehow still got it wrong following this exactly

[@TheIkarus93]

This is not fluid dynamics since there is no analysis of how the fluid behaves and what laws are behind it, this is just calculus.

[@SoMoeYourToe32]

Yeah, I hate this shit.

[@ellieb8627]

prettycoolshii quiz tomorrow... I still have no idea what I'm doing. Lol RIP second quarter AP calc grade

[@Booger564]

Ellie Burggraff I have a quiz in a couple hours!!! wish me luck

[@ellieb8627]

prettycoolshii update: I only got four wrong on my quiz simply because my teacher grades things nicely:)

[@emmabusby9067]

I could barely focus because I was so busy admiring the drawing

[@ky00li0]

THIS WAS SO HELPFUL

[@ahnafkhan8115]

Can we use this rate at which the height is increasing to find when the height will be at its maximum? Any help guys?

[@Nmarryat24]

It should be. He even said pi/12 but wrote pi/2 and continued with it.

[@jessicaschwalb4430]

im sorry but i just gotta say, you allow me to reach unforeseen heights in maf. youre a goddess

[@bisankeneth2710]

What is the given?

[@bisankeneth2710]

Please REPLY A.S.A.P

[@bisankeneth2710]

PLEASEEEEEE!!!!!!

[@been5703]

what if the diameter is not equal

[@Xdrakemanx]

dank

[@Cornbreadddd]

instead of doing (h/2) could you not just put "1"? Because if the relationship is the same, then the height of the water is 2, and the diameter of the surface is 2, meaning the radius is 1. So instead of getting V = pi * h^3 You would get V = (pi * h)/(3) Would that not work as well?

[@allanpeter7700]

did u try it. i think the answer would be different

[@BenjaminFC11]

No, not quite true, although I see where you're going. But the radius of the water within the cone is 1, and the height is two, and that is your relation. 1/2 = r/h. Multiply both sides by h and you get (1 * h)/2 or just h/2.

[@TheWarfare07]

Setting it up always gets me. Reason why I failed the exam twice. Couldn't set it up. 🎃

[@LeinysandMassey4life]

cm^3

[@johnshakal4471]

this is

[@LeinysandMassey4life]

+John Doe two years later

[@johnshakal4471]

finally added you in my 'circles' two years later

[@LeinysandMassey4life]

+John Doe lol. circle life.

[@005Turk]

Hard question!

[@gelatinocyte6270]

What about if the height of the cone is not 4cm, what if it's 6cm?

[@jacobturnblom573]

Making diameter and height of the cone the same made simplifying too easy. None of the problems I am assigned use the same dimension like this example. How would you simplify and pull the derivative when you can't just substitute (h/2) for r?

[@fahadkhalid4507]

how is the r h/2????

[@dcviper985]

because Sal asserts that the height and diameter are the same. And since d=2r, then r=h/2

[@ericm1839]

the ratio of the radius to the height will ALWAYS be the same. so the radius of the largest part of the cone is 2. the ratio is 2/4 (so 1/2) so radius is always 1/2 height

[@someone6949]

you should have kept watching!

[@OmarJoseph]

h(t), cuz it just wasnt confusing enough already.