To find the empty space in the container, we need to subtract the total volume of the tennis balls from the volume of the container.
First, we calculate the volume of one tennis ball using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Given that the radius \( r \) of each tennis ball is 2.5 inches and using 3.14 as the approximation for \( \pi \):
\[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]
Calculating \( (2.5)^3 \):
\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]
Now substituting that into the volume formula:
\[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
Calculating \( \frac{4}{3} \times 3.14 \):
\[ \frac{4}{3} \times 3.14 \approx 4.1867 \]
Now we multiply this result by 15.625:
\[ V \approx 4.1867 \times 15.625 \approx 65.49 \]
Thus, the volume of one tennis ball is approximately 65.49 cubic inches.
Since there are a dozen (12) tennis balls, we multiply the volume of one ball by 12:
\[ \text{Total Volume of Tennis Balls} = 12 \times 65.49 \approx 785.88 \text{ in}^3 \]
Now, we subtract this from the volume of the container to find the empty space:
\[ \text{Empty Space} = \text{Volume of Container} - \text{Total Volume of Tennis Balls} \]
\[ \text{Empty Space} = 3200 - 785.88 \approx 2414.12 \text{ in}^3 \]
Rounding to the nearest whole number gives us:
\[ \text{Empty Space} \approx 2414 \text{ in}^3 \]
Therefore, the empty space in the container is 2,414 cubic inches.
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