To determine how much space in the container is empty, we need to calculate the volume of the dozen tennis balls and subtract that from the total volume of the container.
Step 1: Calculate the volume of one tennis ball
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.
Given that the radius of each tennis ball is 2.5 inches, we substitute \( r = 2.5 \) into the formula. Using \( \pi \approx 3.14 \):
\[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]
Calculating \( (2.5)^3 \):
\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]
Now plug this back into the volume equation:
\[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
Calculating \( \frac{4}{3} \times 3.14 \):
\[ \frac{4}{3} \times 3.14 \approx 4.1867 \]
Now calculate:
\[ V \approx 4.1867 \times 15.625 \approx 65.31 \text{ in}^3 \]
Step 2: Calculate the volume of a dozen tennis balls
Since there are a dozen tennis balls, we multiply the volume of one tennis ball by 12:
\[ \text{Total volume of 12 tennis balls} = 12 \times 65.31 \approx 783.72 \text{ in}^3 \]
Step 3: Calculate the empty space in the container
The total volume of the container is 3,200 in³. To find the empty space, we subtract the total volume of the tennis balls from the volume of the container:
\[ \text{Empty space} = \text{Volume of container} - \text{Total volume of tennis balls} \] \[ \text{Empty space} = 3200 - 783.72 \approx 2416.28 \text{ in}^3 \]
Step 4: Round to the nearest whole number
Rounding \( 2416.28 \) to the nearest whole number gives:
\[ \text{Empty space} \approx 2416 \text{ in}^3 \]
Therefore, the amount of empty space in the container is approximately \( \boxed{2416} \) in³.
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