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?

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