To determine the radius needed for the pole to fit perfectly on the diagonal of the cylinder-shaped container, we can use the Pythagorean theorem.
In this case, we have a right triangle formed by the height of the container, the length of the pole, and the diagonal of the container. The height of the container is 8 ft., and the length of the pole is 10 ft. We can denote the radius of the container as \( r \).
The diagonal \( d \) can be computed using the Pythagorean theorem in a 2D plane, which states:
\[ d = \sqrt{(height)^2 + (diameter)^2} \]
In terms of the radius, the diameter \( D \) is \( 2r \). Therefore:
\[ d = \sqrt{(8)^2 + (2r)^2} \]
We want the length of the pole (10 ft) to equal the diagonal \( d \):
\[ 10 = \sqrt{8^2 + (2r)^2} \]
Now, squaring both sides:
\[ 100 = 64 + 4r^2 \]
Now, we can isolate \( r^2 \):
\[ 100 - 64 = 4r^2 \] \[ 36 = 4r^2 \] \[ 9 = r^2 \]
Taking the square root of both sides:
\[ r = 3 \text{ ft} \]
Thus, the required radius for the pole to fit perfectly on the diagonal is 3.0 ft.