To find the empty space in the container, we first need to calculate the total volume of the tennis balls and then subtract that from the volume of the container.
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Find the volume of one tennis ball:
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius.Given the radius \( r = 2.5 \) inches, we can plug in the values:
\[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]First, calculate \( (2.5)^3 \):
\[ (2.5)^3 = 15.625 \]Now, substituting this back into the volume formula:
\[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
\[ V \approx \frac{4}{3} \times 3.14 \times 15.625 \]
\[ V \approx \frac{4 \times 3.14 \times 15.625}{3} \]Now calculate:
\[ 3.14 \times 15.625 \approx 49.0625 \]
\[ 4 \times 49.0625 \approx 196.25 \]
\[ \frac{196.25}{3} \approx 65.4167 \]Therefore, the volume of one tennis ball is approximately:
\[ V \approx 65.42 , \text{in}^3 \] -
Find the volume of a dozen tennis balls:
Since there are 12 tennis balls:
\[ 12 \times 65.42 \approx 785.04 , \text{in}^3 \] -
Calculate the empty space in the container:
The total volume of the container is 3,200 in³.
Empty space = Total volume - Volume of tennis balls
\[ \text{Empty space} = 3200 - 785.04 \approx 2414.96 \] -
Round to the nearest whole number:
\[ \text{Empty space} \approx 2415 , \text{in}^3 \]
Thus, the amount of empty space in the container is approximately 2415 in³.