AMC 8 2024 Problem 22
AMC8 2024 年第 22 题
圆★★★☆☆
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.
下图为一卷胶带的横截面。胶带直径为4英寸，其中包裹着一个直径为2英寸的圆环。胶带厚度为0.015英寸。请问这卷胶带全部展开大约有多少英寸？请将答案四舍五入到最接近的百位数。
- A.300
- B.600
- C.1200
- D.1500
- E.1800
正确答案 · Correct Answer
B
解析 · Solution
解法1
这卷胶带约 层。为了求出总长度，我们需要求出每个同心圆的平均值并乘以它。由于小圆的直径为2英寸，大圆的直径为4英寸，所以"中间值"（或平均值）为3。因此，平均周长为。将相乘，近似得到(B) 600。
大约有 个"整圈"胶带，平均周长为 这意味着答案是 600。
我们可以将胶带侧面的长度视为一个非常细的矩形，其宽度为0.015英寸。胶带的侧面被包裹成一个环形（两个同心圆之间的阴影区域），这意味着阴影区域的面积等于这个非常细的矩形的面积。阴影区域的面积为，将其除以0.015得到。将近似为3，我们得到最终答案为。
体积守恒法：胶带体积不变，用两种形状各算一次、令它们相等。
摊平后：把整卷胶带摊开叠平，就是一块又长又薄的长方体。三条边为：厚 0.015、长 （要求的胶带总长）、宽 （胶带宽度，即垂直纸面那一维）。体积 。
卷起来时：它是一个「中间挖了洞的圆柱」（截面为内径 2、外径 4 的圆环），高也是 。体积 = 环面积 高 （环面积 ）。
两者相等：，约去 得
如果你没有注意到平均直径是3，你仍然可以按照以下方法解决这个问题。同样地，我们有层胶带。如果我们考虑每层直径均为2，那么长度应为 。如果直径被视为4，那么长度应为800。因此，长度在400到800之间，唯一可能的答案是600。
Solution 1
The roll of tape is about layers thick. In order to find the total length, we have to find the average of each concentric circle and multiply it. 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 . Multiplying gives approximately (B) 600.
There are about "full circles" of tape, and with average circumference of which means the answer is 600.
We can figure out the length of the tape the side of the tape as a really thin rectangle that has a width of 0.015 inches. The side of the tape is wrapped into an annulus(The shaded region between 2 circles with the same center), meaning the area of the shaded region is equal to the area of the really thin rectangle. The area of the shaded region is , and we divide that by 0.015 to get . Approximating to be 3, we get the final answer to be .
Conservation of volume: the tape's volume is the same in two shapes — compute each and equate.
Unrolled: lay the whole roll out flat — a long, thin rectangular box. Its three sides: thickness 0.015, length (the tape length we want), width (the tape's width, into the page). Volume .
Rolled: a cylinder with a hole (cross-section: a ring of inner diameter 2, outer diameter 4), also of height . Volume = ring area height (ring area ).
Equate: ; cancel :
If you cannot notice that the average diameter is 3, you can still solve this problem following method. The same with, we have layers of tape. If we consider every layers with the diameter 2, the length should be . If the diameter is seem as 4, the length should be 800. So, the length is between 400 and 800, the only possible answer is 600.
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