calculus helpp!!!!!

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?

 

 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?

 

 m³/day

Answer 1

Notice that width = 2 * height

and width = 2 * radius

so radius = height, r=h

Set up equation for Volume, then differentiate with respect to time (t). We are given dh/dt = .5 m/day

b.) dw/dt = 0.6 m/day which means dh/dt = 0.3 m/day. Now just substitute new dh/dt into above

Notice that the answers depend on h.