AMC10 2023 B

AMC10 2023 B · Q24

AMC10 2023 B · Q24. It mainly tests Coordinate geometry, Transformations.

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?

由所有可以表示为 $(2u-3w, v+4w)$ 的点的区域的边界的周长是多少，其中 $0\le u\le1$，$0\le v\le1$，且 $0\le w\le1$？

(A) 10\sqrt{3} 10\sqrt{3}

(B) 13 13

(D) 18 18

(E) 16 16

Answer

Correct choice: (E)

正确答案：（E）

Solution

Notice that we are given a parametric form of the region, and $w$ is used in both $x$ and $y$. We first fix $u$ and $v$ to $0$, and graph $(-3w,4w)$ from $0\le w\le1$. When $w$ is $0$, we have the point $(0,0)$, and when $w$ is $1$, we have the point $(-3,4)$. We see that since this is a directly proportional function, we can just connect the dots like this: Now, when we vary $2u$ from $0$ to $2$, this line is translated to the right $2$ units: We know that any points in the region between the line (or rather segment) and its translation satisfy $w$ and $u$, so we shade in the region: We can also shift this quadrilateral one unit up, because of $v$. Thus, this is our figure: The length of the boundary is simply $1+2+5+1+2+5$ ($5$ can be obtained by Pythagorean theorem since we have side lengths $3$ and $4$.). This equals $\boxed{\textbf{(E) }16.}$

注意到我们给出了区域的参数形式，且 $w$ 用于 $x$ 和 $y$。我们首先固定 $u$ 和 $v$ 为 $0$，绘制 $(-3w,4w)$ 从 $0\le w\le1$。当 $w=0$ 时，有点 $(0,0)$，当 $w=1$ 时，有点 $(-3,4)$。我们看到由于这是直接比例函数，我们可以像这样连接点：现在，当我们变化 $2u$ 从 $0$ 到 $2$ 时，这条线向右平移 $2$ 个单位：我们知道线（更确切地说是线段）与其平移之间的区域中的任何点满足 $w$ 和 $u$，所以我们填充区域：我们也可以因为 $v$ 而将这个四边形向上平移一个单位。因此，这就是我们的图形：边界的长度简单地是 $1+2+5+1+2+5$（$5$ 可以通过勾股定理得到，因为边长 $3$ 和 $4$）。这等于 $\boxed{\textbf{(E) }16}$。

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