Question
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 of the statements regarding the sphere with a diameter of 8 cm:
1. **The sphere has a radius of 4 cm.**
- **True**. The radius is half of the diameter. So, the radius = diameter / 2 = 8 cm / 2 = 4 cm.
2. **The sphere has a radius of 16 cm.**
- **False**. As calculated above, the radius is 4 cm.
3. **The diameter’s length is twice the length of the radius.**
- **True**. By definition, the diameter is twice the radius (diameter = 2 * radius).
4. **The radius’s length is twice the length of the diameter.**
- **False**. This is incorrect as the radius is half of the diameter, not twice.
5. **The volume of the sphere is \( \frac{2,048}{3} \pi \) centimeters cubed.**
- **False**. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Here, \( r = 4 \) cm. Thus, \( V = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \) cm³.
6. **The volume of the sphere is \( \frac{256}{3} \pi \) centimeters cubed.**
- **True**. As calculated above, the volume is indeed \( \frac{256}{3} \pi \) cm³.
Thus, the statements that are true are:
- 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 of the diameter. So, the radius = diameter / 2 = 8 cm / 2 = 4 cm.
2. **The sphere has a radius of 16 cm.**
- **False**. As calculated above, the radius is 4 cm.
3. **The diameter’s length is twice the length of the radius.**
- **True**. By definition, the diameter is twice the radius (diameter = 2 * radius).
4. **The radius’s length is twice the length of the diameter.**
- **False**. This is incorrect as the radius is half of the diameter, not twice.
5. **The volume of the sphere is \( \frac{2,048}{3} \pi \) centimeters cubed.**
- **False**. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Here, \( r = 4 \) cm. Thus, \( V = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \) cm³.
6. **The volume of the sphere is \( \frac{256}{3} \pi \) centimeters cubed.**
- **True**. As calculated above, the volume is indeed \( \frac{256}{3} \pi \) cm³.
Thus, the statements that are true are:
- 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.