8
1
2016
24

一道算阴影部分面积的“小学五星题”

首先评价:这道题肯定不是小学水平,至少是初中奥赛(要用三角函数),不过高中生水平应该会做。如果真是一道小学题,那么左下角那一块要涂满。

根据对称性,很容易看出来,如果填上左下角(下图中的ADE区域),那么阴影部分面积会很好求,矩形面积减去两个圆面积除以 2 就行了(这也正是小学题水平,但肯定算不上什么五星题)。

现在问题转化为求左下角(ADE)的面积。先画几条辅助线。

显然有 SADE=SABC-SDEBC,略加运算得到 [tex]S_{ADE}=\frac{25\pi}{4}-(S_{ODEC}-S_{ODC})[/tex]。

为了求 SODEC 和 SODC,有必要知道 ∠DOC。易得 tan∠OCD=tan∠CAB=0.5。为了方便,把 ∠OCD 记作 α,则有 ∠DOC=θ=π-2α。

首先可求出 SODEC:[tex]S_{ODEC}=\frac{25\theta}{2}=\frac{25}{2}(\pi-2\arctan\frac{1}{2})[/tex]。

根据几何关系:OF=5sinα,CF=5cosα,则 SODC = 12.5sin2α = 10。这里需要一定的三角函数知识才可得解。

现在把这些数据代回 SADE 的运算式中即得结果 [tex]10+25\arctan\frac{1}{2}-\frac{25\pi}{4}[/tex],约等于 1.956。

剩下的事情就很好办了,可以得到最终结果——原图中阴影部分面积 [tex]90-\frac{75\pi}{4}-25\arctan\frac{1}{2}[/tex],约等于 19.50。

小学题中不可能出现三角函数,所以这不可能是一道小学题,而是有人故意涂掉了区域ADE出来骗人的……有人似乎通过微积分得解,大概挺复杂,懒得用那个方法算了,不过这个几何办法也挺麻烦的。

Category: 理科 | Tags: | Read Count: 2025
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Jan 14, 2023 01:38:52 PM

This is a "five-star primary school question" to calculate the area of the shaded part. To calculate the area of the shaded part, we need to know the area of the entire rectangle. The area of the entire rectangle is the length times the width. CBD metabolism The length of the rectangle is 8 and the width is 5, so the area of the rectangle is 8 times 5, which is 40.

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