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)

Question

Responses

12.8 ft.
12.8 ft.

6.0 ft.
6.0 ft.

6.4 ft.
6.4 ft.

3.0 ft.

Answer 1

To determine the radius needed for the pole to fit perfectly on the diagonal of the container, we first need to find the diagonal length of the cylinder-shaped container. We can use the Pythagorean theorem for this since the diagonal will create a right triangle with the height and the diameter (which is twice the radius).

Let:

• $h$ = height of the container = 8 ft
• $l$ = length of the pole = 10 ft
• $d$ = diagonal length of the container

We can express the diagonal $d$ using the Pythagorean theorem:

$$d=\sqrt{h^2+(2r{)}^2}$$

Where $r$ is the radius of the cylinder. Since we want the pole to fit perfectly on the diagonal, we can set $d=l$ (the length of the pole).

So, we have:

$$10=\sqrt{8^2+(2r{)}^2}$$

Squaring both sides gives:

$$100=64+(2r{)}^2$$

Subtracting 64 from both sides:

$$36=(2r{)}^2$$

Taking the square root of both sides:

$$2r=6$$

Dividing by 2 to find $r$:

$$r=3$$

Thus, the radius needed for the pole to fit perfectly on the diagonal is $\boxed{{\displaystyle 3}}$ ft.