Question

What is the farthest distance between two points on a cylinder of: HEIGHT 8 and RADIUS 8?

• A. \(8\sqrt{5}\)
• B. \(8\sqrt{2}\)
• C. \(8\sqrt{3}\)
• D. \(16\sqrt{5}\)

Answer 1

The Correct Option is A

To determine the farthest distance between two points on a cylinder, we need to consider one point on the bottom edge of the cylinder and the other on the top edge of the cylinder directly above or opposite to this point.

Given the cylinder's Height (H) = 8 and Radius (R) = 8, one method to find the farthest distance is by using the diagonal of the ‘unrolled’ cylinder.

When the cylinder is ‘unrolled’, it forms a rectangle. The height of this rectangle is the height of the cylinder (8). Its width will be the circumference of the cylinder's base, given by:

\( \text{Circumference} = 2 \pi R = 2 \pi \times 8 = 16 \pi \)

Now, to find the farthest distance:

The farthest possible distance between two points on this rectangle (formed by unrolling) is the diagonal. The formula for the diagonal (d) of a rectangle given its height and width is:

\( d = \sqrt{\text{Height}^2 + \text{Width}^2} \)

Substitute the values:

\( d = \sqrt{8^2 + (16 \pi)^2} \approx \sqrt{64 + 256 \pi^2} \)

Since the options given are in terms of simple square roots and π's complex value makes it unlikely to match any numerical options directly, we consider points on the cylinder's surface. An intuitive approach without further complex calculations is to analyze the logical geometric form of a cylinder.

By considering traditional solutions and typical problems like this, we note the most distant straight-line path could be across the cylinder vertically and curving across its surface to the other side in a path resembling combining both a diameter and height.

For constructive comparison and a likely geometric path within possibilities:

1. The height of 8 adds a straight vertical component possible.

2. The hypothetical surface path from point to point (consider cutting or a stretched most extreme), assuming bending, could incorporate half circumferential influence.

Conclusion: The analysis using allowable interpretations commonly concludes our maximum distance simply as \(\sqrt{4R^2 + H^2} =\sqrt{4(8)^2 + 8^2} = 8\sqrt{5}\), verifying the answer agreement with historical geometry problem outputs.

Thus, the correct answer is:

\( \mathbf{8\sqrt{5}} \)