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A white cylindrical silo has a diameter of $30$ feet and a height of $80$ feet. A red stripe with a horizontal width of $3$ feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

enter image description here

The provided solution says that

The cylinder can be "unwrapped" into a rectangle, to get a stripe that is a parallelogram with base $3$ and height $80$.

However, I don't understand how the height of the stripe can be eighty. The distance from the top of the cylinder to the bottom is $80$ feet as given, so how can a winded stripe that goes around be also $80$ feet? Isn't the shortest distance, $80$ ft, simply from the top to the bottom? How does a winded spiral also become $80$ feet?

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up vote 4 down vote accepted

When the strip is unwound into a parallelogram, the length of the bae is 3, and the height, perpendicular to that base is 80. This is all that matters for the area of a parallelogram. The length of the other side is immaterial.

Note that the number of revolutions the stripe makes is also irrelevant. A strip that made 5 or 10 or 20 revs would have the same area, as long as the horizontal width was the same 3...

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I strongly doubt that the number of revolution doesn't matter. Feel free to draw a picture if the number of revolutions is 4. – Michael Hoppe Jan 21 '14 at 6:50
The number of revolutions must matter. – Claude Leibovici Jan 21 '14 at 6:58
@ClaudeLeibovici: Paradoxically enough, it doesn't. Why ? Because a greater number of revolutions ADDS to the length of the other side, but makes the whole parallelogram THINNER. So, in the end, you give with one hand, but take with the other. Imagine two parallel lines, and a fixed segment of a certain length on each of them. Then let those two equal segments slide freely alongside those two parallel lines, and notice how the area of the parallelogram thus described stays constant, despite the fact that its two other oblique sides gain or loose significantly in size yet its thickness shrinks. – Lucian Jan 21 '14 at 9:48
@Lucian. Thanks for the clarification. It is effcetively very paradoxal. Cheers. – Claude Leibovici Jan 21 '14 at 9:51
@Lucian The number of revolutions does matter if the distance of the parallel lines was constant. I've misread this. – Michael Hoppe Jan 21 '14 at 11:22

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