Question
The volume of the right cylinder = 90π ft^3 with a height of 10 feet.
Check TWO answers that are true.
(2 points)
Responses
The radius of the right cylinder is about 1.7 feet
The radius of the right cylinder is about 1.7 feet
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 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 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 does not have the same volume as the right cylinder with the altitude = the height and the radius the same for both.
The axis of the oblique cylinder is equal to the height of the right cylinder.
The axis of the oblique cylinder is equal to the height of the right cylinder.
The radius of the right cylinder = 3 feet
The radius of the right cylinder = 3 feet
The radius of the right cylinder = 9 feet
Check TWO answers that are true.
(2 points)
Responses
The radius of the right cylinder is about 1.7 feet
The radius of the right cylinder is about 1.7 feet
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 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 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 does not have the same volume as the right cylinder with the altitude = the height and the radius the same for both.
The axis of the oblique cylinder is equal to the height of the right cylinder.
The axis of the oblique cylinder is equal to the height of the right cylinder.
The radius of the right cylinder = 3 feet
The radius of the right cylinder = 3 feet
The radius of the right cylinder = 9 feet
Answers
GPT-4o mini
To determine which statements are true, we first need to find the radius of the right cylinder using the volume formula for a cylinder:
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume
- \( r \) is the radius
- \( h \) is the height
Given that the volume \( V = 90\pi \) ft³ and the height \( h = 10 \) ft, we can set up the equation:
\[
90\pi = \pi r^2 (10)
\]
Dividing both sides by \( \pi \):
\[
90 = 10r^2
\]
Now, divide by 10:
\[
9 = r^2
\]
Taking the square root of both sides:
\[
r = 3 \text{ feet}
\]
Now let's evaluate the statements:
1. **The radius of the right cylinder is about 1.7 feet**
- **False** (the radius is actually 3 feet)
2. **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** (an oblique cylinder can have the same volume as a right cylinder if it has the same radius and height/altitude)
3. **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** (they do have the same volume under the stated conditions)
4. **The axis of the oblique cylinder is equal to the height of the right cylinder.**
- **False** (the axis of an oblique cylinder is not necessarily equal to the height and can differ if they are not perfectly aligned)
5. **The radius of the right cylinder = 3 feet**
- **True** (we calculated the radius to be 3 feet)
6. **The radius of the right cylinder = 9 feet**
- **False** (the radius is 3 feet, not 9)
Therefore, the two true responses are:
- **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.**
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume
- \( r \) is the radius
- \( h \) is the height
Given that the volume \( V = 90\pi \) ft³ and the height \( h = 10 \) ft, we can set up the equation:
\[
90\pi = \pi r^2 (10)
\]
Dividing both sides by \( \pi \):
\[
90 = 10r^2
\]
Now, divide by 10:
\[
9 = r^2
\]
Taking the square root of both sides:
\[
r = 3 \text{ feet}
\]
Now let's evaluate the statements:
1. **The radius of the right cylinder is about 1.7 feet**
- **False** (the radius is actually 3 feet)
2. **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** (an oblique cylinder can have the same volume as a right cylinder if it has the same radius and height/altitude)
3. **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** (they do have the same volume under the stated conditions)
4. **The axis of the oblique cylinder is equal to the height of the right cylinder.**
- **False** (the axis of an oblique cylinder is not necessarily equal to the height and can differ if they are not perfectly aligned)
5. **The radius of the right cylinder = 3 feet**
- **True** (we calculated the radius to be 3 feet)
6. **The radius of the right cylinder = 9 feet**
- **False** (the radius is 3 feet, not 9)
Therefore, the two true responses are:
- **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.**