# Thread: Related Rates about a water tank the shape of a sphere

1. ## Related Rates about a water tank the shape of a sphere

A water tank that is in the shape of a sphere has a radius of 4 meters. When the tank is filled to a depth of
h meters, the volume of water in the tank is

V
=p/3 h^2(12 h).

h is the height of water, measured in meters from the bottom of the tank.

*What values of
h are possible.
*Suppose that the tank is initially empty but that water is poured into the tank at the constant rate of 2 cubic meters per minute. What is the total volume of the tank? How long does it take to fill the tank?

*Determine a formula for the rate of change of the height of thewater with respect to time. Your formula should only depend on
h.

2. The total volume of a sphere is $\frac{4}{3}{\pi}r^{3}$

Therefore, the volume of a sphere with radius 4 meters is $\frac{256\pi}{3}$ cubic meters.

If the water flows in at 2 cubic meters per minute, then it takes:

$\frac{256\pi}{6}=134.04$ minutes.

Can you use related rates to find dh/dt?. Differentiate your formula wrt time.

$\frac{dV}{dt}={\pi}\left(8h-h^{2}\right)\cdot \frac{dh}{dt}$

Solve for dh/dt. You know dV/dt=2