### Home > APCALC > Chapter 12 > Lesson 12.1.5 > Problem12-59

12-59.

If the solid in problem 12-58 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?

Express the main function in terms of $x$, which is the radius.

$x=\Big(\frac{y^2}{16}-1\Big)^{2/3}$

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

$\frac{dV}{dt} = 8\text{ un/min and }\frac{dV}{dt} =\frac{dV}{dh}\cdot\frac{dh}{dt}$

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

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

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