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 \( \boxed{3} \) ft.