A cube is shrinking in size in such a way that the volume of the cube decreases at a constant rate of 30 meters cubed per second. How fast is the side length of the cube changing when the area is 1000 meters squared?

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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.

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