2024 AMC 8 Problems/Problem 22

Problem 22

A roll of tape is $4$ inches in diameter and is wrapped around a ring that is $2$ inches in diameter. A cross section of the tape is shown in the figure below. The tape is $0.015$ inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest $100$ inches.

(A) $300$ (B) $600$ (C) $1200$ (D) $1500$ (E) $1800$

Solution 1

The roll of tape is $1/0.015~66$ layers thick. In order to find the total length, we have to find the average of each concentric circle and multiply it by $66$ . Since the diameter of the small circle is $2$ inches and the diameter of the large one is $4$ inches, the "middle value" is $3$ . Therefore, the average circumference is $3\pi$ . Multiplying $3\pi*66$ gives $(B) \boxed{600}$ .

-ILoveMath31415926535

Solution 2

There are about $\dfrac{1}{0.015}=\dfrac{200}{3}$ "full circles" of tape, and with average circumference of $\dfrac{4+2}{2}\pi=3\pi.$ $\dfrac{200}{3}*3\pi=200\pi,$ which means the answer is $600.$