related rates: find rate of change of volume in filling tank

[sar8364]

A conical tank (with vertex down) is 10 feet across the top and 18 feet deep. As the water flows into the tank, the change is the radius of the water at a rate of 2 feet per minute, find the rate of change of the volume of the water when the radius of the water is 2 feet.

 

[galactus]

This is a case for similar triangles.

If the radius of the water at time t is r and the height at time t is h.

The radius of the cone is 5 and the height is 18.

\(\displaystyle \L\\\frac{r}{h}=\frac{5}{18}\)

\(\displaystyle \L\\h=\frac{18r}{5}\)

Now, sub this into the formula for the volume of a cone, \(\displaystyle V=\frac{1}{3}{\pi}r^{2}h\), and differentiate wrt to t,

Enter in your knowns and you have it.