Module calculus problem

Question

The radius of a right circular cylinder is increasing at the rate of 3 ft/sec, while the height is decreasing at the rate of 6 ft/sec. At what rate is the volume of the cylinder changing when the radius is 15 ft and the height is 10 ft?

1. 30 ft³/sec
2. -900 ft³/sec
3. -450π ft³/sec
4. -900π ft³/sec

Answer 1

The key with these kinds of problems is to write everything that you know down and label everything correctly with the proper signs (+/-).

dr/dt = 3 ft/s

dh/dt = -6 ft/s (note decreasing is negative)

r = 15 ft

h = 10 ft

dV/dt = Rate that volume is changing = ??

First thing that you should be thinking about is... What is the volume formula a right circular cylinder?

V = πr²h

dV/dt = d/dt[πr²h] = π·d/dt[r²h] = π·[r²·dh/dt + h·(2r)·dr/dt] = π·[r²·dh/dt + 2hr·dr/dt]

dV/dt = π·[15²·(-6) + 2(10)(15)(3)] = -450π ft³/s