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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[@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.

[@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 ;]

[@cosmiclens]

@@winterwraith8339 That's good advice man!

[@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

[@abdulazizob]

the drawing is a masterpiece.

[@ericcartman1168]

My first thought

[@eliotwilliams4480]

quite soothing.

[@kinza7549]

holy shit drawing skills on point tho

[@shashankmurthy5668]

+Kinza R Right, dude is good at everything

[@louisehsu8480]

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

[@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..

[@forevereveryours]

Who knew Sal was an artist too, oh dang.

[@dichiethiel7201]

Hannah he is 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

[@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😎😂😂

[@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

[@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

[@arcadianproxy3730]

anyone else internally screaming when he wrote pi/2?

[@camo736]

pi/12 right?

[@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

[@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

[@hunterdixon5845]

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

[@yooncho2112]

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

[@Wolfie020]

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

[@famine784]

Just the question i needed help with THANK YOU!

[@Appl3Muncher]

Thanks khan academy has helped so much.

[@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

[@uchihamadara6024]

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

[@MrAlbashiri]

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

[@k_______p]

Thank you very much!

[@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.

[@ollera18]

you are an artist!

[@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:)

[@djancak]

great explanation, thanks

[@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.

[@zappist751]

GREAT EXPLANATION SIR KHAN

[@kr3wdog06]

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

[@hswhite1]

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

[@Christianturner7]

He corrects it at 10:28 relax.

[@jimcar53]

As usual - Brilliant

[@Nike7548]

Thanks bro lots of help :)

[@behnamasid]

What an amazing answer.

[@thoughtprocess-]

I love you videos. thank you

[@kyuujuuwang]

that was magical

[@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.

[@sarikasabnis2369]

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

[@clueless.2509]

thx

[@Sp0okeh]

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

[@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...😃😯

[@LadyHelenaRavenclaw]

that diagram is a work of art

[@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.

[@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.

[@yashuppot3214]

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

[@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.

[@RobbyBoy167]

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

[@TheMONOPOLY123]

that drawing O.o

[@Frances3654]

very interesting

[@Lauderdalesfinest954754]

Fluid Dymamics..! Cool..

[@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 🤣

[@cosmiclens]

@@NightbladeNottyThat's good to hear man. I have a looong way to go.

[@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

[@raymond-reviews]

Could you guys do related rates for troughs

[@he0k0n41095]

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

[@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

[@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. ◼

[@FlamingAlchemist7]

Did this in Calculus not too long ago

[@FireVortex720]

thnx man

[@xhelixshotx]

The old video on rates of change provides a much shorter and time-saving method to solve this.

[@Crob56]

Something with respect to something and the derivative of something.

[@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.

[@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

[@luiscecenas6681]

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

[@hardikguptaiit-k5028]

Nice video

[@zag5434]

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

[@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.

[@RogerKlauser]

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

[@9starbrite3]

He fixed it after a bit.

[@ismaeeldasti2773]

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

[@lancelovecraft5913]

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

[@Xdrakemanx]

dank

[@ShaMan3D]

How you convert linear motion into circular ???

[@fieldHunter61]

Might as well embrace it kiddies. Got a degree in CS with 4 certifications but still can't find work. I thought I was done with this but here I am reviewing in case I have to become a math teacher. lol

[@RamSharma-zp5fx]

Have you found work yet

[@fieldHunter61]

@@RamSharma-zp5fx Yes! and I love my job as a Java developer! After much perseverance and with the help of a specialized recruiter I didn't become a teacher. Now I'm pursuing a larger vision.

[@emmabusby9067]

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

[@shanebauer8091]

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

[@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).

[@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?

[@HenggaoCai]

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

[@hamonyqueso699]

i have been staring at my question for good 3 hrs now, and now my head hurts. IM DONE

[@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?

[@joshuacoleman714]

he corrects himself if you watch all the way through.

[@flowerpower__]

👍👍👍

[@17puttab]

You meant pi over 12 at 10:13

[@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?

[@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"

[@TheWarfare07]

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

[@jessicaschwalb4430]

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

[@nonchalantd]

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

[@OmarJoseph]

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

[@rewrew5762]

He fixes pi/2 to pi/12 @10:29, don't trip guys LOL

[@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.

[@mingyan1966]

he corrects the mistake later at 10:23

[@gelatinocyte6270]

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

[@waterfairy2011]

I am afraid that I can never figure out what's happening in rate of change Chain rule just freak me out

[@ky00li0]

THIS WAS SO HELPFUL

[@GandTpro95]

thanks. you sound like Joe from family guy lol

[@Cloud_0089]

Oh haha, my mistake. I noticed when I continued watching the video onwards.