Question
C) 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? You must show
all work in order to receive credit. (2 points)
measures 22 cm. How much air can the ball hold if it is pumped to full capacity? You must show
all work in order to receive credit. (2 points)
Answers
GPT-4o mini
To determine how much air the soccer ball can hold when pumped to full capacity, we need to calculate the volume of the ball. A soccer ball is a spherical shape, and we can use the formula for the volume \( V \) of a sphere:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
1. **Find the radius**:
The diameter of the soccer ball is given as 22 cm. The radius \( r \) is half of the diameter:
\[
r = \frac{diameter}{2} = \frac{22 \, \text{cm}}{2} = 11 \, \text{cm}
\]
2. **Plug the radius into the volume formula**:
\[
V = \frac{4}{3} \pi (11 \, \text{cm})^3
\]
3. **Calculate \( r^3 \)**:
\[
r^3 = 11^3 = 11 \times 11 \times 11 = 1331 \, \text{cm}^3
\]
4. **Substitute \( r^3 \) back into the volume formula**:
\[
V = \frac{4}{3} \pi (1331 \, \text{cm}^3)
\]
5. **Calculate \( \frac{4}{3} \times 1331 \)**:
\[
\frac{4}{3} \times 1331 \approx 1774.67 \, \text{cm}^3
\]
6. **Now insert this value back into the equation to find the exact volume**:
\[
V \approx 1774.67 \pi \, \text{cm}^3
\]
If we use \( \pi \approx 3.14 \) for estimation:
\[
V \approx 1774.67 \times 3.14 \approx 5566.67 \, \text{cm}^3
\]
Therefore, the soccer ball can hold approximately **5566.67 cm³** of air when pumped to full capacity.
**Final answer: the soccer ball can hold approximately 5566.67 cm³ of air.**
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
1. **Find the radius**:
The diameter of the soccer ball is given as 22 cm. The radius \( r \) is half of the diameter:
\[
r = \frac{diameter}{2} = \frac{22 \, \text{cm}}{2} = 11 \, \text{cm}
\]
2. **Plug the radius into the volume formula**:
\[
V = \frac{4}{3} \pi (11 \, \text{cm})^3
\]
3. **Calculate \( r^3 \)**:
\[
r^3 = 11^3 = 11 \times 11 \times 11 = 1331 \, \text{cm}^3
\]
4. **Substitute \( r^3 \) back into the volume formula**:
\[
V = \frac{4}{3} \pi (1331 \, \text{cm}^3)
\]
5. **Calculate \( \frac{4}{3} \times 1331 \)**:
\[
\frac{4}{3} \times 1331 \approx 1774.67 \, \text{cm}^3
\]
6. **Now insert this value back into the equation to find the exact volume**:
\[
V \approx 1774.67 \pi \, \text{cm}^3
\]
If we use \( \pi \approx 3.14 \) for estimation:
\[
V \approx 1774.67 \times 3.14 \approx 5566.67 \, \text{cm}^3
\]
Therefore, the soccer ball can hold approximately **5566.67 cm³** of air when pumped to full capacity.
**Final answer: the soccer ball can hold approximately 5566.67 cm³ of air.**