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:

`(dV)/(dt)=3s^2(ds)/(dt) `

(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:

`-30=3(500/3)(ds)/(dt) `

`==> (ds)/(dt)=-3/50 `

So the rate of decrease of the side length is -3/50 meters per second.

**Further Reading**

## We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support

Already a member? Log in here.

Are you a teacher? Sign up now