Calculus- conical pile

Question

Sand is pouring out of a pipe and is forming a conical pile on the ground.The radius of the pile is increasing at a rate of 3 meters per hour.The height is always  of the radius.When the pile has a radius of 15 meters,

a) the Volume is __ cubic meters

b) the rate of change of the VOLUME is __cubic meters per hour..

Answer 1

From geometry we know the Volume of a cone, V is given /by 1/3 x pi x r³ where r is the radius.

Since h = 13 x r, we have V = 13 x pi x r³ and since pi = 15 we have V = 14,625 pi

We use calculus on the second part of the question. It is a related rates type of problem.

We have dV/dh = 1/3 x pi x dr³/dh = 3 x 1/3 x pi x r² = 3 x 1/3 x pi x 15² = 225 pi