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I am pouring water into a conical cup 8cm tall and 6cm across the top. If the volume of the cup at time t is $V(t)$, how fast is the water level ($h$) rising in terms of $V'(t)$?

The solution in the book is:

Take the water volume, given by

$$\frac{1}{3}\pi(\frac{3}{8})^2h^3$$

Then differentiate with respect to $t$:

$$V' = h'\pi(\frac{3}{8})^2h^2$$

Which gives

$$h' = \frac{64V'}{9{\pi}h^2}$$

I did not understand how this differentiation happened, when there was no $t$ in the formula to differentiate! If you differentiate with respect to $h$, though, you get something similar:

$$\dfrac{9{\pi}h^2}{64}$$

But I'm not sure how $V'$ fits into this.

Thanks in advance!

naiveai
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1 Answers1

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To answer in words, making GoodDeeds point more clear. Notice how if you differentiate the whole term with respect to t

$$ V = \frac{1}{3}{\pi}(\frac{3}{8})^2h^3 $$

use chain rule on the h such that the expression becomes

$$ h'*constants *3h^2 $$

where h' is the derivative of h with.

All thats left is to deal with the constants and rearrange.

naiveai
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Jordan Simba
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  • sorry, i'm not used to using this formatting – Jordan Simba Mar 13 '16 at 05:32
  • FTFY, use $\frac$ for fractions – naiveai Mar 13 '16 at 05:35
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    Cheers. But now do you see where the chain rule is used to get to the solution? – Jordan Simba Mar 13 '16 at 05:38
  • Sort of... I though the chain rule could only be used when you have a composite function, like $$h(x) = v(u(x))$$ Then $$h'(x) = u'(x)v'(u(x))$$ But I don't see how that would apply here. – naiveai Mar 13 '16 at 05:40
  • Yes, you're right, compare the problem to what you've just written. The whole term of V is differentiated wrt h then multiplied by the derivative of h wrt to t... i.e chain rule. – Jordan Simba Mar 13 '16 at 05:44
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    So... $$V'(t) = h'(t)V'(h(t))$$ Oh, well duh! I was looking at $h$ like a variable, when in fact you have a composite function! – naiveai Mar 13 '16 at 05:46
  • So to differentiate with respect to a variable doesn't necessarily mean that the variable needs to literally exist in the function... it just means that there has to be a function of that variable somewhere in the function. – naiveai Mar 13 '16 at 05:47
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    Yeah, exactly! Intuitively, we know that the water level h changes with time so its a natural assumption to make that h=h(t). – Jordan Simba Mar 13 '16 at 05:50