A spherical snowball is melting in such a way that its radius is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the radius is 12 cm. (Note the answer is a positive number).

Round your answer to the nearest thousands.

To find the rate at which the volume of the snowball is decreasing, we need to apply related rates. The formula for the volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

Where \( r \) is the radius of the sphere. To find the rate of change of the volume, we differentiate both sides of this equation with respect to time \( t \).

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

Given:
- \( \frac{dr}{dt} = -0.2 \, \text{cm/min} \) (the radius is decreasing, hence the negative sign),
- \( r = 12 \, \text{cm} \).

We can now substitute these values into the equation.

\[
\frac{dV}{dt} = 4 \pi (12^2) (-0.2)
\]

Let's calculate that.

\[
\frac{dV}{dt} = 4 \pi (144) (-0.2)
\]
\[
\frac{dV}{dt} = 4 \pi (28.8)
\]
\[
\frac{dV}{dt} = 115.2 \pi
\]
\[
\frac{dV}{dt} \approx 115.2 \times 3.1416 \approx 361.91 \, \text{cm}^3/\text{min}
\]

Since the problem asks for the rate at which the volume is **decreasing**, the answer is positive. Therefore, the volume is decreasing at a rate of approximately **361.91 cm³/min**.

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Would you like more details or have any questions? Here are some related questions:

1. How would the rate of volume change differ if the radius decreases at a faster rate?
2. What happens to the volume if the radius is halved? 
3. How do related rates work in non-spherical objects?
4. How does the surface area change rate relate to the volume change rate in a sphere?
5. What are the practical applications of related rates in physics or engineering?

**Tip**: When working with related rates, always differentiate implicitly with respect to time.

A spherical snowball is melting in such a way that its radius is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the radius is 12 cm. (Note the answer is a positive number).

Learn how to calculate the rate at which the volume of a spherical snowball decreases as its radius shrinks. In this problem, the radius decreases at 0.2 cm/min, and the volume change rate is found using related rates and differentiation. Suitable for students in Grades 11-12 studying calculus.

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