Question

a spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.4 cm/min. at what rate is the volume of the snowball decreasing when the radius is 11 cm. (note the answer is a positive number).

\\(\text{cm}^3 / \text{min}\\)

hint: the volume of a sphere of radius \\(r\\) is \\(v = \frac{4}{3}\pi r^3\\)

Explanation:

Identify the given rates and variables

Using the Related Rates knowledge point

$$
LATEXBLOCK0
$$

Differentiate the volume equation with respect to time

Using the Related Rates knowledge point

$$ \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} $$

Calculate the rate of change of volume

Using the Related Rates knowledge point

$$
LATEXBLOCK1
$$

Determine the rate of decrease

Using the Related Rates knowledge point

$$ \text{Rate of decrease} = -\frac{dV}{dt} = 193.6\pi\text{ cm}^3/\text{min} $$

Answer:

A spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the radius is 11 cm. (Note the answer is a positive number).

<blank>\(193.6\pi\)</blank> \(\frac{\text{cm}^3}{\text{min}}\)