Question

A machine starts dumping sand at the rate of 35 m3/min, forming a pile in the shape of a cone. The height of the pile is always one and one half the length of the base diameter. After 7 minutes, how fast is the radius increasing? And the height?

Answer 1

We are given with the following given
dV/dt = 35 m3/min
h = (3/2)D

What is asked is
dr/dt at t = 7 min

The volume of a cone is
V = (1/3)πr²h

From the given
h = (3/2)D
since D = 2r
h = 3r

Subsituting
V = πr³

At t = 7 min, the volume is
V = 35(7)
V = 245
and the radius is
245 = πr³
r = 4.27

Differentiating the volume
dV/dt = 3πr² dr/dt
From the given
dV/dt = 35

Subsituting
35 = 3πr² dr/dt
So,
dr/dt = 0.20 m/s
and
dh/dt = 0.60 m/s