AMC8 2024

AMC8 2024 · Q22

AMC8 2024 · Q22. It mainly tests Rounding & estimation, Area & perimeter.

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.

一卷胶带的直径为 $4$ 英寸，缠绕在一个直径为 $2$ 英寸的环上。下图显示了胶带的横截面。胶带厚度为 $0.015$ 英寸。如果完全展开，这卷胶带大约有多长？答案四舍五入到最近的 $100$ 英寸。

(A) 300 300

(B) 600 600

(D) 1500 1500

(E) 1800 1800

Answer

Correct choice: (B)

正确答案：（B）

Solution

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" (or mean) is $3$. Therefore, the average circumference is $3\pi$. Multiplying $3\pi \cdot 66$ gives approximately $(B) \boxed{600}$.

胶卷厚度为 $1/0.015 \approx 66$ 层。为了求总长度，需要计算每层同心圆的平均周长并乘以 $66$。小圆直径 $2$ 英寸，大圆直径 $4$ 英寸，平均值为 $3$。因此平均周长为 $3\pi$。$3\pi \cdot 66$ 约为 $(B) \boxed{600}$。

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