1987 AMC 8 第 22 题

Question

ABCD

是一个矩形，

D

是圆心，

B

在圆上。如果

AD = 4

且

CD = 3

，那么阴影区域面积位于哪两个值之间？

ABCD

is a rectangle,

D

is the center of the circle, and

B

is on the circle. If

AD = 4

and

CD = 3,

then the area of the shaded region is between which two values?

4

和

5

4

and

5

5

和

6

5

and

6

6

和

7

6

and

7

7

和

8

7

and

8

8

和

9

8

and

9

答案：D

知识点：勾股定理扇形

难度评级：1120

小提示：

对角线

DB

是半径；用勾股定理求出它

The diagonal

DB

is the radius; find it with the Pythagorean theorem

大提示：

阴影区域是四分之一圆减去矩形：

\dfrac14 \pi (5)^2 - 3 \cdot 4

The shaded region is a quarter of the circle with the rectangle removed:

\dfrac14 \pi (5)^2 - 3 \cdot 4

解答：

因为

AD = 4

且

CD = 3

，所以对角线

DB = \sqrt{3^2 + 4^2} = 5

，这就是半径。

阴影区域是以

D

为圆心的四分之一圆减去矩形：

\begin{gathered} \dfrac14 \pi (5)^2 - (3)(4) = \dfrac{25\pi}{4} - 12 \\ \approx 19.6 - 12 = 7.6 \end{gathered}\text{。}

这个值位于

7

和

8

之间。

所以正确答案是 D。

Since

AD = 4

and

CD = 3,

the diagonal

DB = \sqrt{3^2 + 4^2} = 5,

which is the radius.

The shaded region is the quarter circle at

D

minus the rectangle:

\begin{gathered} \dfrac14 \pi (5)^2 - (3)(4) = \dfrac{25\pi}{4} - 12 \\ \approx 19.6 - 12 = 7.6. \end{gathered}

This lies between

7

and

8.

Thus, the correct answer is D .

1987 AMC 8 第 22 题

答案：D

难度评级：1120

小提示：

大提示：

解答：

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