3 years ago
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
cleanbandittourcanada Staff answered 3 years ago
cleanbandittourcanada Staff answered 3 years ago
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