To determine the radius required for the pole to fit perfectly on the diagonal inside the cylindrical container, we need to apply the Pythagorean theorem. The pole can fit diagonally across the height and the diameter of the cylinder.
The height of the container (h) is 8 ft and the length of the pole (l) is 10 ft. We need to find the radius (r) of the cylinder.
The formula for the diagonal (d) of the cylinder is given by:
\[ d = \sqrt{(height)^2 + (diameter)^2} \]
The diameter of the cylinder is related to the radius by:
\[ diameter = 2 \cdot r \]
Substituting this into the diagonal formula gives us:
\[ d = \sqrt{(height)^2 + (2r)^2} \]
Now substituting the height and the pole length:
\[ 10 = \sqrt{(8)^2 + (2r)^2} \]
Squaring both sides to eliminate the square root:
\[ 100 = 64 + 4r^2 \]
Subtracting 64 from both sides:
\[ 36 = 4r^2 \]
Now, divide both sides by 4:
\[ 9 = r^2 \]
Taking the square root gives:
\[ r = 3 \]
Thus, the radius needed for the pole to fit perfectly on the diagonal is 3.0 ft.
Final answer: 3.0 ft.