Question

A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. The volume formula for a right circular cone is V=(1/3)pi(r^2)h. A) After 5 minutes, what is the height of the pile? B) After 5 minutes, how fast is the height increasing? C) After 5 minutes, how fast is the base radius increasing? D) After 10 minutes, how fast is the area of the base increasing?

Answer 1

Hints:
Given V(t)=20t \(m^3\)
Volume, V(t) = \(\dfrac{1}{3}\pi R^2 H = \dfrac{1}{3}\pi R^2 4R = \dfrac{4}{3}\pi R^3\)
Equate volumes,
\(20t = \dfrac{4}{3}\pi R^3\)
Solve for R at time t, and height H = 4R.