Question

Wyatt just used his allowance to buy a soccer ball. At full capacity the diameter of the ball measures 22 cm. How much air can the ball hold if it is pumped to full capacity? Show all work

Answer 1

To determine how much air the soccer ball can hold at full capacity, we need to calculate the volume of the ball, which is modeled as a sphere. The formula to calculate the volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

where \( r \) is the radius of the sphere.

First, we need to find the radius of the soccer ball. The diameter of the ball is given as 22 cm, so the radius \( r \) is half of the diameter:

\[
r = \frac{Diameter}{2} = \frac{22 \, \text{cm}}{2} = 11 \, \text{cm}
\]

Now that we have the radius, we can substitute it into the volume formula:

\[
V = \frac{4}{3} \pi (11 \, \text{cm})^3
\]

Next, we calculate \( 11^3 \):

\[
11^3 = 11 \times 11 \times 11 = 121 \times 11 = 1331 \, \text{cm}^3
\]

Now substitute \( 1331 \, \text{cm}^3 \) back into the volume formula:

\[
V = \frac{4}{3} \pi (1331 \, \text{cm}^3)
\]

Now calculate the volume:

\[
V = \frac{4}{3} \times \pi \times 1331
\]

Using \( \pi \approx 3.14159 \):

\[
V = \frac{4}{3} \times 3.14159 \times 1331
\]

Calculating \( \frac{4}{3} \times 3.14159 \) first:

\[
\frac{4}{3} \approx 1.33333
\]
\[
1.33333 \times 3.14159 \approx 4.18879
\]

Now, multiply by \( 1331 \):

\[
V \approx 4.18879 \times 1331 \approx 5567.96 \, \text{cm}^3
\]

Thus, the soccer ball can hold approximately:

\[
\boxed{5568 \, \text{cm}^3}
\]

of air when pumped to full capacity.