We are asked to find the rate at which the side of a cube is decreasing if the volume is decreasing at a steady rate of 30 cubic meters per second at the moment that the surface area of the cube is 100 square meters.
The volume of the cube is given by `V=s^3 ` where s is the side length.
Differentiating both sides with respect to time (t in seconds) we get:
(Here we use the chain rule.)
If the surface area of the cube is 1000 square meters we have:
`1000=6s^2 ==> s^2=500/3 `
With `(dV)/(dt)=-30,s^2=500/3 ` we have:
`==> (ds)/(dt)=-3/50 `
So the rate of decrease of the side length is -3/50 meters per second.