Problem:
A roll of tape is inches in diameter and is wrapped around a ring that is inches in diameter. A cross section of the tape is shown in the figure below. The tape is inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest inches.
Answer Choices:
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Solution:
We have a huge margin of error for this problem so we can freely estimate values. Firstly, we note that the entire roll of tape is -inch thick, and given that a single tape is inches thick, then there are layers of tape in the entire roll.
Then, we must identify an estimate for the circumference of one layer of tape. Near the center, one layer of tape will have a circumference of while the layers near the outer section will have a circumference of about . A good estimate would be to take the average circumference which is .
With this, we can estimate the total length for the entire roll:
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions