To find out how much space in the container is empty, we first need to calculate the volume of one tennis ball and then the total volume of all twelve tennis balls. Then, we will subtract this total volume from the volume of the container.
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Volume of a sphere formula:
\[ V = \frac{4}{3} \pi r^3 \] where \( V \) is the volume and \( r \) is the radius. -
Calculate the volume of one tennis ball:
Given the radius \( r = 2.5 , \text{in} \) and 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 back into the volume formula: \[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
Calculate \( \frac{4}{3} \times 3.14 \): \[ \frac{4}{3} \times 3.14 \approx 4.1867 \]
Now multiply by \( 15.625 \): \[ V \approx 4.1867 \times 15.625 \approx 65.25 , \text{in}^3 \]
So, the volume of one tennis ball is approximately \( 65.25 , \text{in}^3 \).
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Calculate the volume of a dozen tennis balls: \[ \text{Total volume for 12 tennis balls} = 12 \times 65.25 \approx 783 , \text{in}^3 \]
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Calculate empty space in the container: The volume of the container is given as \( 3200 , \text{in}^3 \): \[ \text{Empty space} = \text{Volume of the container} - \text{Total volume of tennis balls} \] \[ \text{Empty space} = 3200 - 783 \approx 2417 , \text{in}^3 \]
Thus, the amount of space in the container that is empty is approximately 2417 in³.