Question
The volume of the right cylinder =90pift^3 with a height of 10 feet. Check TWO answers that are TRUE.
the radius of the right cylinder = 9ft
the oblique cylinder and the right cylinder would have the same volume as long as the height equals the altitude and they have the same radius.
the axis of the oblique cylinder 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 radius of the right cylinder is about 1.7ft
the radius of the right cylinder = 3ft
the radius of the right cylinder = 9ft
the oblique cylinder and the right cylinder would have the same volume as long as the height equals the altitude and they have the same radius.
the axis of the oblique cylinder 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 radius of the right cylinder is about 1.7ft
the radius of the right cylinder = 3ft
Answers
GPT-4o mini
To solve this, we first need to find the radius of the right cylinder with the given volume and height. The volume \( V \) of a right cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- \( V = 90\pi \) ft³,
- \( h = 10 \) ft.
Using the formula:
\[
90\pi = \pi r^2 (10)
\]
We can divide both sides by \( \pi \):
\[
90 = 10 r^2
\]
Then, dividing both sides by 10 gives:
\[
9 = r^2
\]
Taking the square root of both sides yields:
\[
r = 3 \text{ ft}
\]
Now, we analyze the statements provided to see which ones are true:
1. **The radius of the right cylinder = 9ft.**
**False** (the radius is 3 ft).
2. **The oblique cylinder and the right cylinder would have the same volume as long as the height equals the altitude and they have the same radius.**
**True** (this is a property of cylinders).
3. **The axis of the oblique cylinder is equal to the height of the right cylinder.**
**False** (in general, the axis of an oblique cylinder is not equal to the height unless it is a right cylinder).
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 altitude and radius are the same, the volumes are equal).
5. **The radius of the right cylinder is about 1.7ft.**
**False** (the radius is 3 ft).
6. **The radius of the right cylinder = 3ft.**
**True** (this is correct based on our calculation).
The two true statements are:
- The oblique cylinder and the right cylinder would have the same volume as long as the height equals the altitude and they have the same radius.
- The radius of the right cylinder = 3ft.
\[
V = \pi r^2 h
\]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- \( V = 90\pi \) ft³,
- \( h = 10 \) ft.
Using the formula:
\[
90\pi = \pi r^2 (10)
\]
We can divide both sides by \( \pi \):
\[
90 = 10 r^2
\]
Then, dividing both sides by 10 gives:
\[
9 = r^2
\]
Taking the square root of both sides yields:
\[
r = 3 \text{ ft}
\]
Now, we analyze the statements provided to see which ones are true:
1. **The radius of the right cylinder = 9ft.**
**False** (the radius is 3 ft).
2. **The oblique cylinder and the right cylinder would have the same volume as long as the height equals the altitude and they have the same radius.**
**True** (this is a property of cylinders).
3. **The axis of the oblique cylinder is equal to the height of the right cylinder.**
**False** (in general, the axis of an oblique cylinder is not equal to the height unless it is a right cylinder).
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 altitude and radius are the same, the volumes are equal).
5. **The radius of the right cylinder is about 1.7ft.**
**False** (the radius is 3 ft).
6. **The radius of the right cylinder = 3ft.**
**True** (this is correct based on our calculation).
The two true statements are:
- The oblique cylinder and the right cylinder would have the same volume as long as the height equals the altitude and they have the same radius.
- The radius of the right cylinder = 3ft.