Related Rates #2 Using Cones

[@patrickjmt]

Hi all! Wanna help a KZbin education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

[@2doorduude]

If anyone else was confused why the radius is (h/2)^2 when he re-writes the equation, its because you're given that the height = diameter. So half of the diameter = the radius. So when you re-write the equation instead of having r^2 you have (h/2)^2. This part is easy but took me a long time to visualize. Hopefully this helps one of you.

[@brawndon7875]

would it be wrong to just substitute r for with 5 since it is what 10/2 is

[@mahiruhanayo]

thanks bro

[@hxctrae]

I've watched tons and tons of your videos... and I just now realized... You're left-handed! I hope me being left-handed as well gives me an edge on my AP Calculus BC exam TOMORROW ! Thanks for all of your help, Patrick! You've made this an easier year in school for me. I'll spread the word of your legendary skills.

[@gabrielnewson4169]

how did you do?

[@vanessarahimi550]

Just to clear things up. You only use the product rule when you are multiplying 2 VARIABLES. For example, you would use the product rule for (4x)(3x). However, for his example, V=pi/12 * h ^3 is NOT multiplying two variables. In math, Pi is considered a constant, so pi/12 is also a CONSTANT, not a variable. h^3 is a variable, and it is being multiplied by the constant pi/12, so we cannot use the power rule. Also, yes the derivative of a constant is 0, but only if that constant is being added or subtracted. For the example in the video, he is MULTIPLYING a *constant* and a *variable*, the constant is not being added or subtracted! In situations like this, you leave the constant alone, take the derivative of the variable, and then just multiply it by the constant. For example, the derivative of 7x^3 is (7) (3x^2) which when multiplied is equal to 21x^2. Hope this helps

[@aang7505]

Vanessa Rahimi true, but you can still use the product rule and get the correct answer

[@chanthegamer5537]

i think that he used product rule to pi/12 h raised to 3 because pi/12 is attatched to h raised to 3

[@patrickjmt]

@deis7 no, height is not measured in cubic feet. i am not 6 cubic feet tall.

[@Jono.]

I find it hilarious that this problem is for all intensive purposes completely identical to the stuff im working on 8 years later. Thank you for the video, helpful as always!

[@aidangeist3106]

Same thing 13 years later

[@patrickjmt]

what would like done to it?

[@patrickjmt]

@ribbonwing you are taking the derivative with respect to time, hence the dt

[@purpk86]

Patrick, you never cease to amaze me. My friend and I are were stuck on this very exact problem in our homework except that the rate was 30 instead of 20ft/min. I see the light!!! xoxoxox from Los Angeles

[@patrickjmt]

@IslamPlanet glad u like the vids. i already have quite a bit of multivariable stuff, including multiple integrals.

[@jessieskates]

Thank you so much for your videos. I just finished a LATE night study sesh with them, and they helped ever so much! I know I'll be watching more of them as my math career goes on! Thank you again!!

[@patrickjmt]

@KFUPMs2010 that is cause i was using a crappy camera at the time. deal with it!

[@patrickjmt]

@ksingler22 you are very welcome. good luck on the test!

[@patrickjmt]

@FlufeehDarius u know it is

[@sylviao9752]

why was h/2 substituted for r?@patrickjmt

[@MonstarChan3]

h is the diameter. Radius is half of the diameter. Radius is used in the formula for volume

[@CaptainCalculus]

+Sylvia Okonofua It states "diameter and height are always equal" since r=D/2, then r=H/2 also

[@patrickjmt]

happy to help as always

[@patrickjmt]

@beaver2264 implicit differentiation

[@patrickjmt]

@Sadena123 not any more! but aren't we all students of life? : )

[@rampant6055]

polishing up my stuff for final, this helped :)

[@rampant6055]

Rampant thnx

[@patrickjmt]

@Gotenks211 no one is hungry

[@noorzaid6306]

thank you for all of you videos, I will never finish school without them.

[@FrostByte_AC]

My only problem is, I practice these (like I should) and the bitch'll slap a number "e" on the exam. BAM, down the drain. That area of a cone is very tricky too.

[@Wesb1118]

WOW! I didn't even think of using h to represent d, that makes finding r^2 so much simpler! Thank you! I never would have figured out my own problem without thinking of that!

[@angcientrock]

in the beginning, you are given that the height and diameter are equal. By substituting h for r there are less variables to deal with. Because the volume formula uses radius, PJMT had to write it as h/2 (d=2r, r=d/2)

[@anomnomnomous]

I missed the day in class when we were taught this topic. These helped sooo much. Thank you, Patrick!

[@DAsahutube]

This video was great! One thing that I think is an easy throw off it the height divided by 2. Since the height and diameter are equal than the h/2 and radius are equal too. Just to clear things up.

[@dborked]

Great video. You're a lifesaver, I just had a homework problem exactly like this with different numbers. So much easier to understand than my professor. Thanks a bunch!

[@patrickjmt]

ya, you got to use the correct formula! : ) glad the rest helped too!

[@patrickjmt]

@RhaynnxLoveMusic hope you got it correct :)

[@14tim4]

When you differentiate the volume of a cone with respect to time, are you finding the rate in which, say water, is being added from the base to the tip or the tip to the base? When I do a similar question, I take it that water fills from the tip to the base, up-side-down from the example in the video.

[@johnsu913]

:)

[@marioman619]

Thank you! It helps that your handwriting is really neat. Every time my classmates point to their work and explain, I don't see what's going on. Really, thanks. It's MUCH easier than I thought.

[@orangegold1]

@ribbonwing Oh it came from the chain rule.. when he took the derivative it isn't just an exponent shortcut rule... It's chain rule.. you have to multiply but he exponent and the derivative of the function, then the original function, then drop 1 from the power. dh/dt is the derivative of said function.

[@BeastlySin]

@pearlwhite21 because pi/12 is a coefficient and not a constant standing alone. it is like if you are taking the derivative of 2x^2 you get 4x. so in essence the pi/12 and the 2 are the same thing.

[@applessuace]

Every example I find online says something along the lines of "height is always 1/2 radius" or "height and radius are equal." What if I am NOT given this relationship in my problem?

[@linhphan9248]

Hi Patrick! Quick question: Why did you replace r-squared with (h/2)^2 for the volume function? Would you mind explaining that. Thanks!

[@mobilemagnolia]

when you took the derivative, is it the chain rule and the other stuff disappears because it's zero times h cubed?

[@jinheng]

I struggled with a similar question for ages. Lifesaver once again

[@1matth3w1]

hey patrick, at 1:53 is it ok to reduce like terms when taking the derivative? is it possible to reduce the pi/12 and 3h^2 to pi/4 h^2?

[@tseringchhombi2127]

what if the height and the diameter is different

[@Sadena123]

Great videos! Are you a student yourself?

[@sammyhtb]

thanks for helping me graduate high school and helping me in college right now!

[@ribbonwing]

How does that dh/dt get in there at 1:47? Shouldn't it be dh/dv or something?

[@SolarFlare67]

@TryandbeClever The radius (r) is half of the diameter (h in this case), so plug in h/2 for r.

[@beatssports]

why exactly do you leave pi/12 alone when deriving?

[@tahasilat7394]

because it's just a number and the derivative of any number by itself is 0

[@shammaalhosani992]

Why didn't you use the product rule when you were deriving?

[@andrewdeuriarte9147]

Hey Patrick, how come you solved for h and then plugged that h back into the volume formula?

[@Ji23da]

thanks man, much more helpful than in my uni prof's lecture

[@ksingler22]

@patrickJMT this was a wonderful comment. i thumbed it down by accident. i meant to thumb it up. by the way, its 3:30 in the morning and i have a calc test in the morning and when i'm looking up a calc topic on youtube, i specifically look for your name because you are BY FAR the best teacher on here. these videos are so appreciated. thank you, sir.

[@theMiaow]

why is the radius equals h over 2?

[@orangegold1]

@ribbonwing It didn't make sense to me either, and hit explanation doesn't help at all... now I see it though... on both sides of the equation the 1/dt cancels out so you are left with dv = function * dh... that function is the derivative of volume... so it now says dv = dv*dh... dv candles and you are left with the solution for dh = 1, wait never mind.. I just confused myself.. I don't know what he is doing...

[@jeffreyabarca9041]

Why do you plug in the (h/2)^2 for r? I don't see why

[@DThorn619]

H is the diameter so the radius is half of it or H/2

[@LindenKay]

H is actually height but in this problem it is equal to the diameter so you know that radius is half of your diameter or half of your height (height = diameter)

[@RockSmithStudio]

@Gotenks211 because pi is a constant so it is not changed by the derivative.

[@hellothere3163]

This was perfect! Quick and easy to follow, thank you!

[@Prickle]

sorry, dumb question, but how did you come with (h/2)? is that from geometrically because I totally forgot >.

[@H311archer]

I have the highest grade in my calculus class due to your videos. Thanks

[@AceAites]

@LeroyTrey You leave pi/12 alone because it is a constant. Remember if g(x) = k times f(x), then g'(x) = k times f'(x). This was the property he used.

[@howardFL]

after you took the implicit differentiation on both. why you dont need to write V. Like dv/dt*V as the dh/dt h

[@oM4RICa]

Because the derivative of V is 1. So basically it is ( (1) * dv/dt ) But for the H, because the derivative of h^3 is 3h^2 so u get the derivative of h^3 and multiply it by dh/dt..

[@howardFL]

wait how do u get V prime =1

[@oM4RICa]

Isn't the derivative of X equal to 1 ? It's the same for V Take a look at the Derivative Rules: www.mathsisfun.com/calculus/derivatives-rules.html

[@howardFL]

thx i alrdy got it, I don't know what iam thinking at first. thx anyways

[@georgebaidoojr]

Totally Lost. This is like a complete foreign language.

[@wedeldylan]

Math *is* a language, and just like learning French or Spanish, you have to put in time to understand it

[@hahalala60]

why do you still have the pi/12 when you take the derivative on both sides? Isn't it just zero and shouldnt be in the last equation??

[@xTheComedyMoonx]

When taking the derivative why did pi/12 get left alone?

[@michaelbartonmdb3]

because it is a constant

[@keym3955]

Since the pi/12 is times(*) to the h^3, it is left alone but if it were plus(+) then it would be 0 in derivative form. :)

[@dbpanther94]

can you calculate the rate of increase of my Calc grade as the # of your vids I watch increases?

[@OrangeBerry42]

So I put in 20/[(pi/12)(3x)(100)] into my calculator, multiple times and I got 22918.31181 not .254648. What am I doing wrong?

[@POWERUPE]

There should be nothing wrong as 20/(300pi/21) = 20/78.5398 = 0.254648

[@robbietree1920]

i put 4 + 4 in my calculator but i didnt get right answer what am i doin rong?

[@ouran909]

wow thank you for this! Definitely feeling better about rates.

[@abney317]

@TryandbeClever it's the radius r = diameter/2

[@yogesh892]

thanks so much this helped me with a problem i was stuck on for ever!

[@TheBlazeofRazgriz]

@pearlwhite21 the pi/12 came from the derivative of 1/3pi(h/2)^2

[@patrickjmt]

well, the constant just gets carried along when taking the derivative...

[@BrendaPortugal]

why is the diameter the same as height?

[@TheCombineAssassin]

How did you get h^3?

[@LOLxUnique]

My mind clicked, thanks for tutorial. Much love from australia :)

[@Dreamchaser1231]

how did he find the radius to be (h/2)^2?

[@ryane8633]

H is the diameter so r is half the diameter

[@gameplaydude2011]

just a simple question, why did he leave the pie/12 alone?

[@Unv12sL]

how did you get h/2 again? on 2nd step?

[@patrickjmt]

@dbpanther94 i would need more data first : ) which i would love to have actually; i would love to test a group of students who did watch regularly vs those who did not

[@patrickjmt]

ok, glad u figured it out! : )

[@RussianCosmonauts]

Why do you leave the pi over 12 alone?

[@wtakphysics]

we are doing problems like this one consistently in my high school non ap calculus class

[@scoobynicker]

How did r get replaced by h/2?

[@VenessaWallace93]

I think its because h represents the diameter and since the radius is half of the diameter then it would ok to replace r with h/2

[@dolphinsatsunset1]

Venessa Wallace Actually the problem stated that the diameter and height are always equal which is why h = 2r so then by algebra you obtain, r = h/2.

[@sylviao9752]

dolphinsatsunset1 thanks, I just understood that from your comment

[@dolphinsatsunset1]

Sylvia Okonofua you're welcome. :)

[@patrickjmt]

sure, and i do take the derivative!

[@MrOmar4159]

good looking out bro, you're the homie for this

[@Crystalkatana44Falco45]

Why was is h/2 squared?

[@Ricenugget]

@junior10tu10papa H is the diameter and you want the radius value, which is the diameter (h) divided by 2.

[@khangvl94]

The diameter and the height are always equal. Radius is half of the diameter, which is also half of height. Also, it's much easier to solve for 1 variable (dh/dt) than 2 (dr/dt and dh/dt).

[@dia8449]

thank god for u patrickjmt

[@yhxr1997]

First and foremost, dv/dt = dv/dh * dh/dt dv/dt = 20 dv/dh = f'(pi/12*h^3) h = 10 Our ultimate goal is to find dh/dt Showing this: dv/dt = dv/dh * dh/dt can help the audience to have a clearer idea of what exactly you are doing. Nonetheless, I hope this helps!

[@hudakhaled6746]

You're good but in my country the solution is not like that and i can understand it could you help me please?

[@Waranle]

Thank you Patrick :)

[@Sukii]

why did r become h/2??

[@12121994]

It's 5:30 in the morning and my final for calculus is 3 hours away...

[@sammyhtb]

love ur vids. please do a vid on related rates with cylinders! thx

[@Juxtaroberto]

Because the r in the formula for the volume is squared, but since we know r=h/2, it's h/2 we square.

[@RussianCosmonauts]

you just got me confused a little for saying we leave the pi over 12 alone, nevermind. Thanks for replying

[@jarrettcastpn8351]

Why didn't the pi/12 cancel when you took the derivative with respect to time

[@RogerKlauser]

It's a constant so you could pull that out when taking the derivative.

[@68hinduhammers]

thank keep doing what you are doing.

[@jombatlag9193]

just an hours away to our FINALS !! hope it will work Patrick :P

[@who8mahbacon]

@Gotenks211 because pi is just a number

[@fahadfardan]

why you don't film this with more quality it seems old (1990) take a nice camera and continue >>> however ,thank you very match

[@meganhewitt92]

THANK YOU!

[@IamPromise15]

Why did he find the radius?

[@IamPromise15]

Nvm I got it

[@haydenkarkainen1167]

20/(pi/12)*300 simplifies to 4/(5pi)