Question
Consider this sphere with a diameter of 5 m.
A sphere with diameter 5 meters.
What can be concluded about the sphere? Check all that apply.
The sphere has a radius of 10 cm.
The diameter measure is substituted into the formula to find the volume.
The radius is half the diameter.
The formula to apply is V = four-thirds B h
The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height.
A sphere with diameter 5 meters.
What can be concluded about the sphere? Check all that apply.
The sphere has a radius of 10 cm.
The diameter measure is substituted into the formula to find the volume.
The radius is half the diameter.
The formula to apply is V = four-thirds B h
The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height.
Answers
GPT-4o mini
Given that the sphere has a diameter of 5 meters, let's analyze the statements provided.
1. **The sphere has a radius of 10 cm.**
**Incorrect.** The radius of the sphere is half of the diameter. Since the diameter is 5 meters, the radius is \( \frac{5 \text{ m}}{2} = 2.5 \text{ m} \), which is 250 cm, not 10 cm.
2. **The diameter measure is substituted into the formula to find the volume.**
**Partially correct.** The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). To use this formula, you need to convert the diameter into the radius (i.e., divide by 2). The diameter itself is not directly substituted.
3. **The radius is half the diameter.**
**Correct.** By definition, the radius is always half of the diameter. With a diameter of 5 meters, the radius is indeed \( 2.5 \) meters.
4. **The formula to apply is V = four-thirds B h.**
**Incorrect.** This formula is not applicable to a sphere. The correct formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \).
5. **The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height.**
**Correct.** The volume of the sphere is indeed \( \frac{2}{3} \) of that of a cylinder with the same radius (2.5 m) and height (5 m). The volume of the cylinder is calculated as \( V = \pi r^2 h \), which when you compare to the volume of the sphere leads to the relation mentioned.
In summary, the conclusions are:
- The sphere has a radius of 10 cm: **False**
- The diameter measure is substituted into the formula to find the volume: **Partially correct**
- The radius is half the diameter: **True**
- The formula to apply is V = four-thirds B h: **False**
- The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height: **True**
1. **The sphere has a radius of 10 cm.**
**Incorrect.** The radius of the sphere is half of the diameter. Since the diameter is 5 meters, the radius is \( \frac{5 \text{ m}}{2} = 2.5 \text{ m} \), which is 250 cm, not 10 cm.
2. **The diameter measure is substituted into the formula to find the volume.**
**Partially correct.** The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). To use this formula, you need to convert the diameter into the radius (i.e., divide by 2). The diameter itself is not directly substituted.
3. **The radius is half the diameter.**
**Correct.** By definition, the radius is always half of the diameter. With a diameter of 5 meters, the radius is indeed \( 2.5 \) meters.
4. **The formula to apply is V = four-thirds B h.**
**Incorrect.** This formula is not applicable to a sphere. The correct formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \).
5. **The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height.**
**Correct.** The volume of the sphere is indeed \( \frac{2}{3} \) of that of a cylinder with the same radius (2.5 m) and height (5 m). The volume of the cylinder is calculated as \( V = \pi r^2 h \), which when you compare to the volume of the sphere leads to the relation mentioned.
In summary, the conclusions are:
- The sphere has a radius of 10 cm: **False**
- The diameter measure is substituted into the formula to find the volume: **Partially correct**
- The radius is half the diameter: **True**
- The formula to apply is V = four-thirds B h: **False**
- The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height: **True**