Question

If a snowball melts so that its surface area decreases at a rate of 3 cm2/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 9 cm. (Round your answer to three decimal places.)

Answer 1

The diameter decreases at a rate of 0.053 cm/min.

Explanation:

Surface area of an snowball

The surface area of an snowball has the following equation:

`S_(a) = \pi d^2`

In which d is the diameter.

Implicit differentiation:

To solve this question, we differentiate the equation for the surface area implictly, in function of t. So

`(dS_(a))/(dt) = 2d\pi(dd)/(dt)`

Surface area decreases at a rate of 3 cm2/min

This means that
`(dS_(a))/(dt) = -3`

Tind the rate (in cm/min) at which the diameter decreases when the diameter is 9 cm.

This is
`(dd)/(dt)` when
`d = 9`. So

`(dS_(a))/(dt) = 2d\pi(dd)/(dt)`

`-3 = 2*9\pi(dd)/(dt)`

`(dd)/(dt) = -(3)/(18\pi)`

`(dd)/(dt) = -0.053`

The diameter decreases at a rate of 0.053 cm/min.