A ping pong ball has the inner radius of 24.8 mm and the white material's thickness is 0.4 mm. Use differentials to estimate the volume of the material needed to make 100 ping pong balls. Round your answer to the nearest whole number.



We are tasked with estimating the volume of the material needed to make 100 ping pong balls, using differentials. The volume of a sphere is given by:

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

Where:
- \( r \) is the radius of the sphere.
- \( V \) is the volume.

For the ping pong ball:
- The **inner radius** \( r_{\text{inner}} = 24.8 \, \text{mm} \)
- The **thickness** of the material \( t = 0.4 \, \text{mm} \), which means the **outer radius** \( r_{\text{outer}} = 24.8 \, \text{mm} + 0.4 \, \text{mm} = 25.2 \, \text{mm} \).

We are interested in estimating the volume of the thin shell (the material) using differentials. To do this, we can calculate the change in volume (\( dV \)) corresponding to a small change in radius (\( dr = 0.4 \, \text{mm} \)).

The differential of the volume \( V = \frac{4}{3} \pi r^3 \) is:

\[
dV = 4 \pi r^2 \, dr
\]

Substituting \( r = 24.8 \, \text{mm} \) and \( dr = 0.4 \, \text{mm} \) into the formula:

\[
dV = 4 \pi (24.8)^2 \times 0.4
\]

Now, let's calculate the volume of the material for one ball and then multiply it by 100 to find the total volume for 100 balls.

The estimated volume of the material needed to make 100 ping pong balls is approximately **309,153 mm³** (or 309 cubic centimeters), rounded to the nearest whole number.

Would you like further details on the calculation or clarification on any part?

Here are 5 related questions to expand your understanding:
1. How does the thickness of the material affect the volume of the ping pong ball?
2. How would the result change if the thickness were doubled?
3. What is the volume of a hollow ping pong ball with the same dimensions?
4. How would you calculate the exact volume using the difference of two spheres' volumes?
5. How can the differential method be applied to other shapes?

**Tip:** When solving for small changes in quantities, differentials are a powerful tool to estimate the result without needing to calculate exact values.

A ping pong ball has the inner radius of 24.8 mm and the white material's thickness is 0.4 mm. Use differentials to estimate the volume of the material needed to make 100 ping pong balls. Round your answer to the nearest whole number.

Estimate the volume of material needed for 100 ping pong balls using differential calculus. The inner radius is 24.8 mm, and the material thickness is 0.4 mm. This problem illustrates how to apply the formula for the volume of a sphere and differentials for estimation, suitable for Grades 11-12.

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