To find the empty space in the container, we first need to calculate the total volume occupied by the dozen tennis balls and then subtract that from the volume of the container.
-
Volume of one tennis ball: The volume \( V \) of a sphere can be calculated using the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.
Given that the radius \( r \) of each tennis ball is 2.5 inches, we can substitute this value into the formula: \[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \]
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 \( \frac{4}{3} \times 3.14 \): \[ \frac{4}{3} \times 3.14 \approx 4.18667 \]
Now, calculate the volume: \[ V \approx 4.18667 \times 15.625 \approx 65.36 \text{ in}^3 \] Thus, the volume of one tennis ball is approximately 65.36 in³.
-
Volume of a dozen tennis balls: Since there are a dozen tennis balls (12 balls), we multiply the volume of one ball by 12: \[ \text{Total Volume of 12 tennis balls} = 12 \times 65.36 \approx 784.32 \text{ in}^3 \]
-
Volume of the container: The volume of the container is given as 3,200 in³.
-
Empty space in the container: To find the empty space, subtract the volume occupied by the tennis balls from the volume of the container: \[ \text{Empty space} = \text{Volume of container} - \text{Volume of 12 tennis balls} \] \[ \text{Empty space} = 3200 - 784.32 \approx 2415.68 \text{ in}^3 \]
-
Round to the nearest whole number: The empty space rounded to the nearest whole number is: \[ \approx 2416 \text{ in}^3 \]
Thus, the amount of empty space in the container is approximately 2416 in³.