Question

If a snowball melts so that its surface area decreases at a rate of 5 cm2/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 9 cm

Answer 1

0.0221 cm/min

Explanation:

Surface area of a snowball:

An snowball has a spherical format.

The surface area of an sphere is given by:

`A = 4\pi r^2`

In which r is the radius(half the diameter). So in function of the diameter, we have that:

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

Implicit derivative:

To solve this question, we need to find the implicit derivative of A in function of t.

The variables are A and d, so:

`(dA)/(dt) = 8\pi d (dd)/(dt)`

Its surface area decreases at a rate of 5 cm2/min

This means that
`(dA)/(dt) = -5`

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

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

`(dA)/(dt) = 8\pi d (dd)/(dt)`

`-5 = 8\pi*9 (dd)/(dt)`

`(dd)/(dt) = -(5)/(72\pi)`

`(dd)/(dt) = -0.0221`

This means that the diameter decreases at a rate of 0.0221 cm/min when the diameter is 9 cm.