To find the difference in volumes of the youth and adult softballs, we first need to calculate the volumes of both spheres using the formula for the volume of a sphere, which is:
\[ V = \frac{4}{3} \pi r^3 \]
Step 1: Calculate the radius for both softballs
- Youth softball: Diameter = 3.5 in, so radius \( r = \frac{3.5}{2} = 1.75 \) in
- Adult softball: Diameter = 3.8 in, so radius \( r = \frac{3.8}{2} = 1.9 \) in
Step 2: Calculate the volume of the youth softball
Using \( \pi \approx 3.14 \):
\[ V_{youth} = \frac{4}{3} \pi (1.75)^3 \]
Calculating \( (1.75)^3 \):
\[ (1.75)^3 = 5.359375 \]
Now substituting back into the volume formula:
\[ V_{youth} = \frac{4}{3} \times 3.14 \times 5.359375 \]
Calculating:
\[ V_{youth} \approx \frac{4 \times 3.14 \times 5.359375}{3} \approx \frac{67.066625}{3} \approx 22.35554 \text{ in}^3 \]
Step 3: Calculate the volume of the adult softball
\[ V_{adult} = \frac{4}{3} \pi (1.9)^3 \]
Calculating \( (1.9)^3 \):
\[ (1.9)^3 = 6.859 \]
Now substituting back into the volume formula:
\[ V_{adult} = \frac{4}{3} \times 3.14 \times 6.859 \]
Calculating:
\[ V_{adult} \approx \frac{4 \times 3.14 \times 6.859}{3} \approx \frac{86.02712}{3} \approx 28.67571 \text{ in}^3 \]
Step 4: Find the difference in volumes
\[ \text{Difference} = V_{adult} - V_{youth} = 28.67571 - 22.35554 \approx 6.32017 \text{ in}^3 \]
Rounding to the nearest tenth:
\[ \text{Difference} \approx 6.3 \text{ in}^3 \]
Conclusion
The approximate difference in their volumes is 6.3 in³.