Question
Suppose △ABC
with A(0,0)
, B(3,5)
, and C(0,5)
is rotated about side CB¯¯¯¯¯¯¯¯
. Identify the three-dimensional object formed.(1 point)
Responses
a cone with a diameter of 5 units and a height of 3 units
a cone with a diameter of 5 units and a height of 3 units
a cone with a diameter of 6 units and a height of 5 units
a cone with a diameter of 6 units and a height of 5 units
a cone with a height of 3 units and a diameter of 10 units
a cone with a height of 3 units and a diameter of 10 units
a cone with a height of 5 units and a diameter 3 units
with A(0,0)
, B(3,5)
, and C(0,5)
is rotated about side CB¯¯¯¯¯¯¯¯
. Identify the three-dimensional object formed.(1 point)
Responses
a cone with a diameter of 5 units and a height of 3 units
a cone with a diameter of 5 units and a height of 3 units
a cone with a diameter of 6 units and a height of 5 units
a cone with a diameter of 6 units and a height of 5 units
a cone with a height of 3 units and a diameter of 10 units
a cone with a height of 3 units and a diameter of 10 units
a cone with a height of 5 units and a diameter 3 units
Answers
GPT-4o mini
To understand the three-dimensional object formed when triangle ABC is rotated about side \( CB \), let's first analyze the setup.
Given the points:
- \( A(0, 0) \)
- \( B(3, 5) \)
- \( C(0, 5) \)
1. **Identify the length of side \( CB \)**:
- From point \( C(0, 5) \) to point \( B(3, 5) \), we see that both points have the same y-coordinate (5), meaning \( CB \) is a horizontal line.
- The length of \( CB \) can be calculated as the difference in the x-coordinates of \( C \) and \( B\):
\[ |x_B - x_C| = |3 - 0| = 3 \text{ units} \]
2. **Determine the height**:
- The height of the cone formed during rotation will be the distance from point \( A(0, 0) \) to line \( CB \).
- The y-coordinate of point \( A \) is \( 0 \), while the y-coordinate of line \( CB \) is \( 5 \), so the height of the cone is \( 5 \) units.
3. **Determine the base radius**:
- The radius of the cone's base will be \( 1.5 \) units because the base of the cone is created by the rotation of point \( A \) around line \( CB \), which is halfway of the length \( CB \):
\[ \text{Base radius} = \frac{\text{Length of } CB}{2} = \frac{3}{2} = 1.5 \text{ units} \]
- Therefore, the diameter of the base is \( 3 \) units, because diameter is \( 2 \times \text{radius} \).
Based on this analysis:
- Height = \( 5 \) units
- Diameter = \( 3 \) units
Thus, the correct answer is: **a cone with a height of 5 units and a diameter of 3 units**.
Given the points:
- \( A(0, 0) \)
- \( B(3, 5) \)
- \( C(0, 5) \)
1. **Identify the length of side \( CB \)**:
- From point \( C(0, 5) \) to point \( B(3, 5) \), we see that both points have the same y-coordinate (5), meaning \( CB \) is a horizontal line.
- The length of \( CB \) can be calculated as the difference in the x-coordinates of \( C \) and \( B\):
\[ |x_B - x_C| = |3 - 0| = 3 \text{ units} \]
2. **Determine the height**:
- The height of the cone formed during rotation will be the distance from point \( A(0, 0) \) to line \( CB \).
- The y-coordinate of point \( A \) is \( 0 \), while the y-coordinate of line \( CB \) is \( 5 \), so the height of the cone is \( 5 \) units.
3. **Determine the base radius**:
- The radius of the cone's base will be \( 1.5 \) units because the base of the cone is created by the rotation of point \( A \) around line \( CB \), which is halfway of the length \( CB \):
\[ \text{Base radius} = \frac{\text{Length of } CB}{2} = \frac{3}{2} = 1.5 \text{ units} \]
- Therefore, the diameter of the base is \( 3 \) units, because diameter is \( 2 \times \text{radius} \).
Based on this analysis:
- Height = \( 5 \) units
- Diameter = \( 3 \) units
Thus, the correct answer is: **a cone with a height of 5 units and a diameter of 3 units**.