AMC10 2011 B

AMC10 2011 B · Q14

AMC10 2011 B · Q14. It mainly tests Quadratic equations, Area & perimeter.

A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter of the parking lot?

一个矩形停车场对角线长 25 米，面积 168 平方米。停车场的周长是多少米？

(A) 52 52

(B) 58 58

(D) 68 68

(E) 70 70

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let $x$ and $y$ be the length and width of the parking lot, respectively. Then $xy = 168$ and $x^2 + y^2 = 25^2$. Note that $$(x+y)^2 = x^2 + y^2 + 2xy = 25^2 + 2\cdot 168 = 961.$$ Hence the perimeter is $2(x+y) = 2\cdot \sqrt{961} = 62$. Note that the dimensions of the parking lot are $7$ and $24$ meters.

答案（C）：设 $x$ 和 $y$ 分别为停车场的长和宽。则 $xy=168$ 且 $x^2+y^2=25^2$。注意 $$(x+y)^2=x^2+y^2+2xy=25^2+2\cdot168=961。$$ 因此周长为 $2(x+y)=2\cdot\sqrt{961}=62$。注意停车场的尺寸为 $7$ 米和 $24$ 米。

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