calculus urgent help!! i got the answers its wrong pleasse help with detail

Question

A volcanic cone spews fine particles of rock which land in the surrounding area and cause the size of the cone to increase.

(a) Suppose that a volcanologist takes measurements that show that when the cone is 100 meters wide and 50 meters tall, its height is increasing at 0.5 meters/day. If the assumption is made that the cone will retain its height-width proportion, what is the rate at which the volume of rock is being released by the volcano?

 

1250pi m³/day

(b) Suppose that the volcanologist takes more measurements and realizes the original assumption was poorly made, because the width is increasing at 0.6 meters/day. What would be the new finding for the rate at which the volume of rock is being released by the volcano?

 i got 10,471.98 (but it said its wrong!!) helpppp pleaseee!!!

 m³/day

Answer 1

I assume you only need help with part b?

We are told that dh/dt=0.5 and then dw/dt=0.6 where w=width. But, w=2*r, so dr/dt=0.3.

Now, V=1/3*pi*r^2*h. So, dV/dt=1/3*pi*(2r*dr/dt*h+r^2*dh/dt) by the chain rule.

Also, we know that r=50 and h=50, so plugging into the equation above we get: dV/dt=1/3*pi*(2*50*0.3*50+50^2*0.5)=1/3*pi*(1500+1250)=1/3*pi*2750=2879.79