Question
A sphere has a diameter of 8 cm. Which statements about the sphere are true? Check all that apply.
The sphere has a radius of 4 cm.
The sphere has a radius of 16 cm.
The diameter’s length is twice the length of the radius.
The radius’s length is twice the length of the diameter.
The volume of the sphere is StartFraction 2,048 Over 3 EndFraction pi centimeters cubed.
The volume of the sphere is StartFraction 256 Over 3 EndFraction pi centimeters cubed.
The sphere has a radius of 4 cm.
The sphere has a radius of 16 cm.
The diameter’s length is twice the length of the radius.
The radius’s length is twice the length of the diameter.
The volume of the sphere is StartFraction 2,048 Over 3 EndFraction pi centimeters cubed.
The volume of the sphere is StartFraction 256 Over 3 EndFraction pi centimeters cubed.
Answers
GPT-4o mini
Let's analyze each statement regarding the sphere with a diameter of 8 cm.
1. **The sphere has a radius of 4 cm.**
- **True**. The radius is half the diameter. So, the radius \( r = \frac{8}{2} = 4 \) cm.
2. **The sphere has a radius of 16 cm.**
- **False**. As mentioned, the radius is 4 cm.
3. **The diameter’s length is twice the length of the radius.**
- **True**. The diameter is indeed twice the radius. \( 8 \, \text{cm} = 2 \times 4 \, \text{cm} \).
4. **The radius’s length is twice the length of the diameter.**
- **False**. The radius is half the diameter, not double it.
5. **The volume of the sphere is \( \frac{2,048}{3} \pi \) centimeters cubed.**
- **False**. The volume \( V \) of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \). Substituting \( r = 4 \):
\[
V = \frac{4}{3} \pi (4^3) = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3.
\]
6. **The volume of the sphere is \( \frac{256}{3} \pi \) centimeters cubed.**
- **True**. As calculated above, the volume is \( \frac{256}{3} \pi \, \text{cm}^3 \).
### Summary of True Statements:
- The sphere has a radius of 4 cm.
- The diameter’s length is twice the length of the radius.
- The volume of the sphere is \( \frac{256}{3} \pi \) centimeters cubed.
1. **The sphere has a radius of 4 cm.**
- **True**. The radius is half the diameter. So, the radius \( r = \frac{8}{2} = 4 \) cm.
2. **The sphere has a radius of 16 cm.**
- **False**. As mentioned, the radius is 4 cm.
3. **The diameter’s length is twice the length of the radius.**
- **True**. The diameter is indeed twice the radius. \( 8 \, \text{cm} = 2 \times 4 \, \text{cm} \).
4. **The radius’s length is twice the length of the diameter.**
- **False**. The radius is half the diameter, not double it.
5. **The volume of the sphere is \( \frac{2,048}{3} \pi \) centimeters cubed.**
- **False**. The volume \( V \) of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \). Substituting \( r = 4 \):
\[
V = \frac{4}{3} \pi (4^3) = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3.
\]
6. **The volume of the sphere is \( \frac{256}{3} \pi \) centimeters cubed.**
- **True**. As calculated above, the volume is \( \frac{256}{3} \pi \, \text{cm}^3 \).
### Summary of True Statements:
- The sphere has a radius of 4 cm.
- The diameter’s length is twice the length of the radius.
- The volume of the sphere is \( \frac{256}{3} \pi \) centimeters cubed.