To find how much space in the container is empty, we first need to determine the total volume occupied by the tennis balls.
The volume \( V \) of a sphere (which is the shape of a tennis ball) is calculated using the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.
Given that the radius of each tennis ball is \( r = 2.5 \) inches, we can substitute this value into the formula along with \( \pi \approx 3.14 \):
\[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]
First, we calculate \( (2.5)^3 \): \[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]
Now, we can calculate the volume of one tennis ball: \[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
Calculating \( \frac{4}{3} \times 3.14 \): \[ \frac{4}{3} \times 3.14 \approx 4.18667 \]
Now multiply this by \( 15.625 \): \[ V \approx 4.18667 \times 15.625 \approx 65.37 \text{ in}^3 \]
This is the volume of one tennis ball. Since there are a dozen (12) tennis balls in the container, we multiply the volume of one ball by 12:
\[ \text{Total volume of 12 balls} = 12 \times 65.37 \approx 784.44 \text{ in}^3 \]
Next, we find the empty space in the container by subtracting the total volume occupied by the tennis balls from the volume of the container:
\[ \text{Empty space} = \text{Volume of container} - \text{Volume of balls} \] \[ \text{Empty space} = 3200 - 784.44 \approx 2415.56 \text{ in}^3 \]
Finally, we round this to the nearest whole number: \[ \text{Empty space} \approx 2416 \text{ in}^3 \]
Thus, the amount of empty space in the container is approximately 2416 in³.
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