AMC8 2007

AMC8 2007 · Q16

AMC8 2007 · Q16. It mainly tests Exponents & radicals, Circle theorems.

Amanda Reckonwith draws five circles with radii 1, 2, 3, 4 and 5. Then for each circle she plots the point (C, A), where C is its circumference and A is its area. Which of the following could be her graph?

Amanda Reckonwith 画了五个半径分别为 1、2、3、4 和 5 的圆。然后对于每个圆，她绘制点 (C, A)，其中 C 是其周长，A 是其面积。以下哪一个可能是她的图像？

(A)

(B)

(C)

(D)

(E)

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) The circumferences of circles with radii 1 through 5 are $2\pi$, $4\pi$, $6\pi$, $8\pi$ and $10\pi$, respectively. Their areas are, respectively, $\pi$, $4\pi$, $9\pi$, $16\pi$ and $25\pi$. The points $(2\pi,\pi)$, $(4\pi,4\pi)$, $(6\pi,9\pi)$, $(8\pi,16\pi)$ and $(10\pi,25\pi)$ are graphed in (A). It is the only graph of an increasing quadratic function, called a parabola.

(A) 半径为 1 到 5 的圆的周长分别是 $2\pi$、$4\pi$、$6\pi$、$8\pi$ 和 $10\pi$。它们的面积分别是 $\pi$、$4\pi$、$9\pi$、$16\pi$ 和 $25\pi$。点 $(2\pi,\pi)$、$(4\pi,4\pi)$、$(6\pi,9\pi)$、$(8\pi,16\pi)$ 和 $(10\pi,25\pi)$ 被绘制在 (A) 中。这是一个递增的二次函数的唯一图像，称为抛物线。

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