https://file.aiquickdraw.com/mathbot/file/1726285756640wtmj448x.jpg

We are given the radius \( r \) of a melting snowball as a function of time:

\[
r(t) = 60 - 2.5t
\]

where \( r \) is in centimeters and \( t \) is the time in hours. The snowball is spherical, so the formula for its volume is:

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

We need to find a formula for the volume \( V \) as a function of time \( t \). To do this, we substitute the expression for \( r(t) \) into the volume formula:

\[
V(t) = \frac{4}{3} \pi \left( 60 - 2.5t \right)^3
\]

This expression gives the volume of the snowball in cubic centimeters as a function of time in hours.

Would you like me to expand and simplify this formula further, or solve for specific values? 

Here are some follow-up questions for a deeper understanding:

1. What happens to the volume when \( t = 0 \)?
2. At what time will the snowball completely melt, i.e., when \( r(t) = 0 \)?
3. How does the rate of change of volume behave as the snowball melts?
4. What would the formula be if the melting rate was different, say \( r(t) = 60 - kt \)?
5. How can we find the instantaneous rate of change of the volume with respect to time?

**Tip:** When working with functions that involve powers and time, understanding how derivatives work can help find rates of change, which is important in physical problems like this one!

The radius, r, in centimeters, of a melting snowball is given by r = 60 - 2.5t, where t is the time in hours. The snowball is spherical with volume V = 4/3πr³ cm³. Find a formula for V as a function of time.

Learn how to find the volume of a melting snowball given its radius as a function of time. This problem covers key mathematical concepts such as algebraic functions, spherical geometry, and applying the volume formula for a sphere. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation