Question

一卷胶带直径为 $4$ 英寸，缠绕在直径为 $2$ 英寸的圆环上。胶带截面如下图所示。胶带厚 $0.015$ 英寸。若把胶带完全展开，它大约有多长？答案四舍五入到最接近的 $100$ 英寸整数倍。

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.

$300$

$600$

$1200$

$1500$

$1800$

答案：B

知识点：圆周长估算

难度评级：1580

小提示：

胶带占据一个外半径 $2$ 、内半径 $1$ 的圆环

The tape occupies an annulus with outer radius $2$ and inner radius $1$

大提示：

胶带截面积等于厚度乘以总长度

Area of tape cross-section equals thickness times total length

视频讲解：

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文字解答：

胶带占据一个外半径为 $2$ 英寸、内半径为 $1$ 英寸的圆环，所以它的截面积为 $\pi(2^2-1^2)=3\pi\text{。}$

胶带展开后，同一个截面成为厚 $0.015$ 英寸、长 $L$ 英寸的长方形。因此 $0.015L=3\pi\text{，}$ 所以 $L=200\pi\approx628$ 英寸。四舍五入到最接近的 $100$ 英寸整数倍，胶带长约 $600$ 英寸。

所以正确答案是 B。

The tape occupies an annulus with outer radius $2$ inches and inner radius $1$ inch, so its cross-sectional area is $\pi(2^2-1^2)=3\pi.$

When unrolled, this same cross section is a rectangle of thickness $0.015$ inches and length $L.$ Thus $0.015L=3\pi,$ so $L=200\pi\approx628$ inches. Rounded to the nearest $100$ inches, this is $600$ inches.

Thus, B is the correct answer.

2024 AMC 8 第 22 题