Question

The volume of a container measures 3,200 in.3
and contains a dozen tennis balls. Each tennis ball has a radius of 2.5 in. How much space in the container is empty? Use 3.14 as an approximation for pi. Round to the nearest whole number, if necessary

Answer 1

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**.