Question
If the diameter of a small red beach ball is 8 inches, then the cone with the same radius and a height of 8 inches would fit into the beach ball _______times and the volume of the cone would be about ______in^3.
2
89
134
1/3
3
1
268
536
2/3
2
89
134
1/3
3
1
268
536
2/3
Answers
Answered by
GPT-4o mini
To solve this problem, we need to find the volume of the cone and the volume of the sphere (beach ball), and then compare those.
1. **Finding the volume of the cone:**
- The formula for the volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
- For the cone, we are given:
- Radius \( r = \frac{8 \text{ inches}}{2} = 4 \text{ inches} \)
- Height \( h = 8 \text{ inches} \)
- Plugging in the values:
\[
V = \frac{1}{3} \pi (4)^2 (8) = \frac{1}{3} \pi (16)(8) = \frac{1}{3} \pi (128) = \frac{128}{3} \pi \text{ cubic inches}
\]
- To calculate the numerical volume:
\[
V \approx \frac{128}{3} \times 3.14 \approx 134.04 \text{ in}^3 \quad \text{(using } \pi \approx 3.14\text{)}
\]
2. **Finding the volume of the beach ball (sphere):**
- The formula for the volume of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
- For the beach ball, the radius \( r = 4 \text{ inches} \) (same as the cone):
- Plugging in the values:
\[
V = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cubic inches}
\]
- To calculate the numerical volume:
\[
V \approx \frac{256}{3} \times 3.14 \approx 268.08 \text{ in}^3 \quad \text{(again using } \pi \approx 3.14\text{)}
\]
3. **Calculating how many times the cone volume fits into the beach ball volume:**
- We need to divide the volume of the sphere by the volume of the cone:
\[
\text{Number of cones} = \frac{\text{Volume of sphere}}{\text{Volume of cone}} = \frac{\frac{256}{3} \pi}{\frac{128}{3} \pi} = \frac{256/3}{128/3} = \frac{256}{128} = 2
\]
Thus, the cone with the given dimensions would fit into the beach ball **2 times**, and the volume of the cone is approximately **134 in³**.
Putting the answers together:
- It fits into the beach ball: **2 times**
- The volume of the cone: **134 in³**
1. **Finding the volume of the cone:**
- The formula for the volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
- For the cone, we are given:
- Radius \( r = \frac{8 \text{ inches}}{2} = 4 \text{ inches} \)
- Height \( h = 8 \text{ inches} \)
- Plugging in the values:
\[
V = \frac{1}{3} \pi (4)^2 (8) = \frac{1}{3} \pi (16)(8) = \frac{1}{3} \pi (128) = \frac{128}{3} \pi \text{ cubic inches}
\]
- To calculate the numerical volume:
\[
V \approx \frac{128}{3} \times 3.14 \approx 134.04 \text{ in}^3 \quad \text{(using } \pi \approx 3.14\text{)}
\]
2. **Finding the volume of the beach ball (sphere):**
- The formula for the volume of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
- For the beach ball, the radius \( r = 4 \text{ inches} \) (same as the cone):
- Plugging in the values:
\[
V = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cubic inches}
\]
- To calculate the numerical volume:
\[
V \approx \frac{256}{3} \times 3.14 \approx 268.08 \text{ in}^3 \quad \text{(again using } \pi \approx 3.14\text{)}
\]
3. **Calculating how many times the cone volume fits into the beach ball volume:**
- We need to divide the volume of the sphere by the volume of the cone:
\[
\text{Number of cones} = \frac{\text{Volume of sphere}}{\text{Volume of cone}} = \frac{\frac{256}{3} \pi}{\frac{128}{3} \pi} = \frac{256/3}{128/3} = \frac{256}{128} = 2
\]
Thus, the cone with the given dimensions would fit into the beach ball **2 times**, and the volume of the cone is approximately **134 in³**.
Putting the answers together:
- It fits into the beach ball: **2 times**
- The volume of the cone: **134 in³**