To determine the radius required for the pole to fit perfectly on the diagonal of the cylinder-shaped container, we can consider the container's cross-section, which is a circle, and the diagonal measurement which can be represented as the hypotenuse of a right triangle formed by the height and the diameter of the container.
Here’s a step-by-step breakdown of the problem:
-
Dimensions of the container:
- Height (\(h\)) = 8 ft.
- The length of the pole (\(L\)) = 10 ft.
-
Understanding the diagonal in relation to the container: The pole can fit perfectly diagonal in the space created by the height of the container and the diameter of the cylinder. To find this diagonal, we will make use of the Pythagorean theorem. The diagonal \(d\) is the hypotenuse of a right triangle where one leg is the height \(h\) and the other leg is the diameter of the cylinder.
-
Relating diameter and radius: Let \(r\) be the radius of the cylinder. Then, the diameter \(D\) can be expressed as: \[ D = 2r \]
-
Using the Pythagorean theorem: According to the Pythagorean theorem: \[ d = \sqrt{h^2 + D^2} = \sqrt{h^2 + (2r)^2} \] For our specific problem, substituting the known height: \[ d = \sqrt{8^2 + (2r)^2} \] \[ d = \sqrt{64 + 4r^2} \]
-
Set diagonal equal to the length of the pole: We want the diagonal \(d\) to equal the length of the pole \(L\), which is 10 ft. Thus, we can set up the equation: \[ \sqrt{64 + 4r^2} = 10 \]
-
Square both sides to eliminate the square root: \[ 64 + 4r^2 = 100 \]
-
Solve for \(r^2\): \[ 4r^2 = 100 - 64 \] \[ 4r^2 = 36 \] \[ r^2 = 9 \] \[ r = 3 \]
-
Final result: The radius \(r\) needed for the pole to fit perfectly on the diagonal is 3 ft. Rounded to the nearest tenth, the answer remains 3.0 ft.