Question

If a snowball melts so that its surface area decreases at a rate of 5 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm. (Give your answer correct to 4 decimal places.)

Answer 1

Answer:

The diameter decreases at the rate of 0.0796 cm/min when the diameter is 10 cm.

Step-by-step explanation:

A snowball is spherical, so it's area is given by the following formula:

$A = 4\pi r^{2}$

The radius is half the diameter so:

$A = 4\pi (\frac{d}{2})^{2}$

$A = 4\pi \frac{d^{2}}{4}$

$A = \pi d^{2}$

If a snowball melts so that its surface area decreases at a rate of 5 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm.

We have to find $\frac{dd}{dt}$ when $\frac{dA}{dt} = -5, d = 10$

$A = \pi d^{2}$

Applying implicit differentiation:

We have to variables(A and d), so:

$\frac{dA}{dt} = 2\pi d \frac{dd}{dt}$

$-5 = 2\pi (10) \frac{dd}{dt}$

$\frac{dd}{dt} = -\frac{5}{20\pi}$

$\frac{dd}{dt} = -0.0796$

The diameter decreases at the rate of 0.0796 cm/min when the diameter is 10 cm.