Question

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?

Answer 1

the proportion states that
h=w/2=r
where r is the radius of the cone
Volume of a cone is\[V=(1/3)\pi r^2h=(1/3) \pi h^3\]where we made the substitution h=r
now just take the derivative w/respect to time:\[dV/dt=\pi h^2 (dh/dt)\]plug in and solve!