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 scenario, we have a cylinder whose height is 8 ft and whose diagonal runs from one point on the edge of the base to the top at the opposite edge. The length of the pole, which we'll consider as the hypotenuse of a right triangle formed by the height of the container and the diameter of the container, is equal to 10 ft.
- Let \( r \) be the radius of the base of the cylinder. The diameter of the base would then be \( 2r \).
- The right triangle consists of the height \( (h = 8 , \text{ft}) \) as one leg, the diameter \( (d = 2r) \) as the other leg, and the hypotenuse as the length of the pole \( (10 , \text{ft}) \).
Using the Pythagorean theorem: \[ h^2 + d^2 = \text{length}^2 \] Substituting for \( h \) and \( d \): \[ 8^2 + (2r)^2 = 10^2 \] Simplifying this: \[ 64 + 4r^2 = 100 \] Subtracting 64 from both sides: \[ 4r^2 = 36 \] Dividing by 4: \[ r^2 = 9 \] Taking the square root: \[ r = 3 \]
Thus, the required radius for the pole to fit perfectly on the diagonal is \( \boxed{3.0} \) ft.