Question

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.(1 point)
in.3

Answer 1

To find 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 a dozen tennis balls.

1. **Calculate the volume of one tennis ball:**

The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]

Given that the radius \( r \) of each tennis ball is 2.5 inches, we can substitute the values into the formula.

Using \( \pi \approx 3.14 \):
\[
V = \frac{4}{3} \times 3.14 \times (2.5)^3
\]

First, calculate \( (2.5)^3 \):
\[
(2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625
\]

Now plug this value back into the volume formula:
\[
V = \frac{4}{3} \times 3.14 \times 15.625
\]

Calculate \( \frac{4}{3} \times 3.14 \):
\[
\frac{4}{3} \times 3.14 \approx 4.18667\ (approximately)
\]

Now multiplying by 15.625:
\[
V \approx 4.18667 \times 15.625 \approx 65.267
\]

So, the volume of one tennis ball is approximately:
\[
V \approx 65.27\ (rounding\ to\ 2\ decimal\ places)
\]

2. **Calculate the total volume of a dozen (12) tennis balls:**
\[
\text{Total Volume} = 12 \times V \approx 12 \times 65.27 \approx 783.24
\]

3. **Determine the empty space in the container:**

The total volume of the container is given as 3200 cubic inches. Therefore, the empty space is:
\[
\text{Empty Space} = \text{Volume of Container} - \text{Total Volume of Tennis Balls}
\]
\[
\text{Empty Space} = 3200 - 783.24 \approx 2416.76
\]

Rounding this to the nearest whole number gives us:
\[
\text{Empty Space} \approx 2417\ \text{in}^3
\]

Thus, the amount of empty space in the container is approximately **2417 in³**.