Question

AL Athletics makes poles for pole vaulting. They use cylinder-shaped containers for shipping. If the company is mailing a pole that is 10 ft. in length in a container that has a height of 8 ft., what radius is needed for the pole to fit perfectly on the diagonal? Round the answer to the nearest tenth. (1 point)
Responses

3.0 ft.
3.0 ft.

6.4 ft.
6.4 ft.

12.8 ft.
12.8 ft.

6.0 ft.

Answer 1

To fit the pole diagonally in the cylindrical container, we need to calculate the diagonal distance across the base of the cylinder and ensure that the pole's length can fit within this diagonal.

1. **Find the height of the cylinder**: The container is 8 ft tall.
2. **Find the length of the pole**: The pole is 10 ft long.

3. **Determine the relationship between the dimensions**:
The pole will fit perfectly diagonally if the diagonal length of the cylinder’s base is at least as long as the length of the pole.

4. **Using the Pythagorean theorem**, the relationship can be expressed as follows:
\[
d = \sqrt{h^2 + (2r)^2}
\]
where \(d\) is the diagonal, \(h\) is the height of the cylinder, and \(r\) is the radius of the cylinder.

5. **Set up the equation**:
Set the diagonal equal to the length of the pole:
\[
\sqrt{8^2 + (2r)^2} = 10
\]

6. **Square both sides to eliminate the square root**:
\[
64 + (2r)^2 = 100
\]

7. **Rearranging gives us**:
\[
(2r)^2 = 100 - 64
\]
\[
(2r)^2 = 36
\]
\[
2r = 6
\]

8. **Solving for \(r\)**:
\[
r = \frac{6}{2} = 3 \text{ ft}
\]

Thus, the radius needed for the pole to fit perfectly on the diagonal is **3.0 ft**.

So the correct response is:
**3.0 ft.**