Question
The volume of the right cylinder = 90 pi ft3 with a height of 10 feet. Check answer that are true: (The radius of the right cylinder is about 1.7 feet; The radius of the right cylinder = 9 feet; The axis of the oblique is equal to the height of the right cylinder; The oblique cylinder does not have the same volume as the right cylinder with the altitude = the height and the radius the same for both; The oblique cylinder and the right cylinder would have the same volume as long as the height = the altitude and they have the same radius; The radius of the right cylinder = 3 feet)
Answers
GPT-4o mini
To find the radius of the right cylinder, we can use the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume of the cylinder
- \( r \) is the radius
- \( h \) is the height
Given that the volume \( V = 90\pi \) ft³ and the height \( h = 10 \) ft, we can rearrange the formula to solve for the radius \( r \):
\[
90\pi = \pi r^2 (10)
\]
Dividing both sides by \( \pi \):
\[
90 = 10r^2
\]
Now, divide both sides by 10:
\[
9 = r^2
\]
Taking the square root of both sides gives us:
\[
r = 3 \text{ feet}
\]
Now let's evaluate the statements provided:
1. **The radius of the right cylinder is about 1.7 feet**: **False** (The radius is 3 feet).
2. **The radius of the right cylinder = 9 feet**: **False** (The radius is 3 feet).
3. **The axis of the oblique is equal to the height of the right cylinder**: This statement is not well-defined without additional context about the oblique cylinder, so we cannot determine if it's true or false without more information.
4. **The oblique cylinder does not have the same volume as the right cylinder with the altitude = the height and the radius the same for both**: **False**. If the oblique cylinder has the same radius and the same height (altitude), it would have the same volume as the right cylinder.
5. **The oblique cylinder and the right cylinder would have the same volume as long as the height = the altitude and they have the same radius**: **True** (If the height and radius are the same, the volumes will be equal).
6. **The radius of the right cylinder = 3 feet**: **True** (We calculated it and found it to be 3 feet).
In summary:
- True: 5, 6
- False: 1, 2, 4
- Statement 3 cannot be evaluated without additional information.
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume of the cylinder
- \( r \) is the radius
- \( h \) is the height
Given that the volume \( V = 90\pi \) ft³ and the height \( h = 10 \) ft, we can rearrange the formula to solve for the radius \( r \):
\[
90\pi = \pi r^2 (10)
\]
Dividing both sides by \( \pi \):
\[
90 = 10r^2
\]
Now, divide both sides by 10:
\[
9 = r^2
\]
Taking the square root of both sides gives us:
\[
r = 3 \text{ feet}
\]
Now let's evaluate the statements provided:
1. **The radius of the right cylinder is about 1.7 feet**: **False** (The radius is 3 feet).
2. **The radius of the right cylinder = 9 feet**: **False** (The radius is 3 feet).
3. **The axis of the oblique is equal to the height of the right cylinder**: This statement is not well-defined without additional context about the oblique cylinder, so we cannot determine if it's true or false without more information.
4. **The oblique cylinder does not have the same volume as the right cylinder with the altitude = the height and the radius the same for both**: **False**. If the oblique cylinder has the same radius and the same height (altitude), it would have the same volume as the right cylinder.
5. **The oblique cylinder and the right cylinder would have the same volume as long as the height = the altitude and they have the same radius**: **True** (If the height and radius are the same, the volumes will be equal).
6. **The radius of the right cylinder = 3 feet**: **True** (We calculated it and found it to be 3 feet).
In summary:
- True: 5, 6
- False: 1, 2, 4
- Statement 3 cannot be evaluated without additional information.