Sand falls from a conveyor belt at the rate of 10 m^3/min onto the top of s conical pile. The height of the pile is always three-eighths of the base diameter. How fast is the radius changing when the pile is 4 m high? Give your answer in cm/min,

1 Answers

Best Answer

Step 1

Diameter is 2r.

Given: rate of volume increase and relationship between h an r.

We're looking for dh/dt when h=4. Plug in h=4 to find r at the same moment for use later.

dV/dt=10

h=3/8(2r)=3/4r

(4)=3/4r

r=4/3*4=16/3

Step 2

V=1/3 pi r^2 h

=1/3 pi r^2*(3r)/4

=1/4 pi r^3

dV/dt=3/4 pi r^2 dr/dt

Step 3

Plug in the values for r and dV/dt to solve.

3/4 pi r^2 dr/dt=dV/dt

3/4 pi (16/3)^2 dr/dt=10

dr/dt=10*3/(64pi)

=15/(32 pi) m/min * 100cm/1m

=1500/(32 pi) cm/min=375/(8 pi) cm/min

~14.92 cm/min