CPM Homework Help : CALC Problem 12-60

The solid in problem 12-59 is a storage tank and liquid is being added at a rate of $8$ cubic units per minute. How fast is the level of the liquid in the tank rising when the level is $4$ units above the lowest point of the tank?

Hint:

Graph the functions.
The lowest point is at $y = 4$ .
The height is $y$ . The radius is $x$ .
The volume will be $πx^2dy$ .

Steps:

Express the main function in terms of $x$ , which is the radius.
$V=\int_4^{h+4}\pi\Big(\frac{y^2}{16}-1\Big)^{4/3}dy$

$\frac{dV}{dt}=\frac{d}{dt}\int_4^{h+4}\pi\Big(\frac{y^2}{16}-1\Big)^{4/3}dy$

$8=\pi\Big(\frac{(h+4)^2}{16}-1\Big)^{4/3}\cdot\frac{dh}{dt}$

Let $h = 4$ and solve for $dh/dt$ .