Hello. This is a related rates problem.
You are given the following:
DV/dt = 40cm^3, when r = 10cm
You want: dr/dt:
Dv/dt = DV/dr . dr/dt
-> 40 = 4. pi. r^2 . (dr/dt)
-> 40 = 4 . pi . (100) . (dr/dt)
-> dr/dt = 40 / ((400) . (pi))
-> dr/dt = 1 / ((10) . (pi) )
-> dr/dt = 0.031830988
To test:
DV/dt = DV/dr . (dr/dt)
-> DV/dt = 400 pi . (0.031830988)
-DV/dt = 40
Therefore dr/dt does, in fact = 0.031830988
Best of luck.