To determine the amount of empty space in the container, we first need to calculate the volume of one tennis ball and then the total volume of all the tennis balls. After that, we can subtract the total volume of the tennis balls from the volume of the container.
- 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.
Given the radius of the tennis ball is \( r = 2.5 \) in and using \( \pi \approx 3.14 \): \[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]
Calculating \( (2.5)^3 \): \[ (2.5)^3 = 15.625 \]
Now, substitute \( 15.625 \) back into the volume formula: \[ V = \frac{4}{3} \times 3.14 \times 15.625 \]
Calculating \( \frac{4}{3} \times 3.14 \): \[ \frac{4}{3} \times 3.14 \approx 4.1867 \]
Now multiply this by \( 15.625 \): \[ V \approx 4.1867 \times 15.625 \approx 65.375 \]
So, the volume of one tennis ball is approximately \( 65.375 \) in³.
-
Calculate the volume of a dozen tennis balls:
Since there are 12 balls: \[ \text{Total volume of 12 balls} = 12 \times 65.375 \approx 784.5 \text{ in}^3 \] -
Determine the empty space in the container:
The volume of the container is \( 3,200 \) in³. Now we subtract the volume of the tennis balls from the container's volume to find the empty space: \[ \text{Empty space} = \text{Volume of container} - \text{Total volume of tennis balls} \] \[ \text{Empty space} \approx 3200 - 784.5 \approx 2415.5 \text{ in}^3 \] -
Round to the nearest whole number:
The empty space is approximately \( 2416 \) in³ when rounded to the nearest whole number.
Final Answer: 2416 in³