Question

Carl 决定给他的长方形花园围篱笆。他买了 $20$ 根篱笆桩，在四个角各放一根，其余沿花园边均匀放置，相邻桩之间正好相距 $4$ 码。花园较长的一边（包括角上的桩）桩数是较短一边（包括角上的桩）的两倍。Carl 的花园面积是多少平方码？

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?

$256$

$336$

$384$

$448$

$512$

答案：B

知识点：植树问题方程组矩形周长

难度评级：1440

小提示：

设较短边有 $x$ 根桩，较长边有 $2x$ 根桩；四个角桩被共享。

Let the shorter side have $x$ posts and the longer side $2x$ posts; the four corner posts are shared

大提示：

总桩数 $2x+2(2x)-4=20$ ；一条边有 $k$ 根桩时长度为 $(k-1)\cdot4$ 码。

Total posts $2x+2(2x)-4=20;$ a side with $k$ posts spans $(k-1)\cdot4$ yards

解答：

设较短边有 $x$ 根桩，则较长边有 $2x$ 根。四个角都被重复计算一次，所以 $2x+2(2x)-4=20$ ，得 $x=4$ 。较短边有 $4$ 根桩，长度为 $(4-1)\cdot4=12$ 码；较长边有 $8$ 根桩，长度为 $(8-1)\cdot4=28$ 码。面积为 $12\cdot28=336$ 。

所以正确答案是 B。

Let the shorter side have $x$ posts, so the longer side has $2x.$ Counting all posts and subtracting the four corners counted twice, $2x+2(2x)-4=20,$ giving $x=4.$ The shorter side has $4$ posts, or $(4-1)\cdot4=12$ yards, and the longer side has $8$ posts, or $(8-1)\cdot4=28$ yards. The area is $12\cdot28=336.$

Thus, the correct answer is B.

2016 AMC 12B 第 9 题

答案：B

难度评级：1440

小提示：

大提示：

解答：

其他年份的第 9 题