Question

If a snowball melts so that its surface area decreases at a rate of 9 cm2/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 10 cm. (Round your answer to three decimal places.)

Answer 1

The diameter decreases at a rate of 0.143cm/min when it is of 10 cm.

Explanation:

Surface area of an snowball:

An snowball has spherical format. The surface area of an sphere is given by:

`S = d^2\pi`

In which d is the diameter of the sphere.

In this question:

We need to differentiate S implicitly in function of time. So

`(dS)/(dt) = 2d\pi(dd)/(dt)`

Surface area decreases at a rate of 9 cm2/min

This means that
`(dS)/(dt) = -9`

At which the diameter decreases when the diameter is 10 cm?

This is
`(dd)/(dt)` when
`d = 10`. So

`(dS)/(dt) = 2d\pi(dd)/(dt)`

`-9 = 2(10)\pi(dd)/(dt)`

`(dd)/(dt) = -(9)/(20\pi)`

`(dd)/(dt) = -0.143`

Area in cm², so diameter in cm.

The diameter decreases at a rate of 0.143cm/min when it is of 10 cm.