To find the amount of empty space in the container, we first need to calculate the total volume of the tennis balls and then subtract that from the total volume of the container.
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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. In this case, the radius \( r \) is 2.5 inches. Using \( \pi \approx 3.14 \):
\[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]
First, calculate \( (2.5)^3 \):
\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]
Now substitute this value into the volume formula:
\[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
Calculate \( \frac{4}{3} \times 3.14 \):
\[ \frac{4}{3} \times 3.14 \approx 4.18667 \]
Now multiply by \( 15.625 \):
\[ V \approx 4.18667 \times 15.625 \approx 65.280625 \]
So, the volume of one tennis ball is approximately \( 65.28 \) cubic inches.
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Calculate the total volume of a dozen tennis balls: Since there are 12 tennis balls:
\[ \text{Total Volume of 12 balls} = 12 \times 65.28 \approx 783.36 , \text{in}^3 \]
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Calculate the empty space in the container: The volume of the container is given as 3,200 cubic inches. Therefore, the empty space is:
\[ \text{Empty Space} = \text{Volume of Container} - \text{Total Volume of 12 balls} \]
\[ \text{Empty Space} = 3200 - 783.36 \approx 2416.64 , \text{in}^3 \]
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Round to the nearest whole number: The final step is to round \( 2416.64 \) to the nearest whole number, which is \( 2417 \).
Therefore, the amount of empty space in the container is approximately 2417 cubic inches.
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