Question

If a snowball melts so that its surface area decreases at a rate of 3 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm.

Answer 1

The diameter decreases at a rate of -0.0477 cm/min when it is 10 cm.

Explanation:

Surface area of a snowball:

A snowball has a spheric format, which means that it's surface area is given by:

`A = 4\pi r^2`

In which r is the radius, which is half the diameter. In function of the diameter, the area is given by:

`A = 4\pi((d)/(2))^2 = \pi d^2`

Solving the question:

To solve this question, we have to implicitly derivate the area in function of t. So

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

Snowball melts so that its surface area decreases at a rate of 3 cm2/min

This means that
`(dA)/(dt) = -3`

Find the rate at which the diameter decreases when the diameter is 10 cm.

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

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

`-3 = 20\pi(dd)/(dt)`

`(dd)/(dt) = -(3)/(20\pi)`

`(dd)/(dt) = -0.0477`

The diameter decreases at a rate of -0.0477 cm/min when it is 10 cm.