题目

Problem

The diagram shows a sector of a circle with centre $O$ and radius $r$ cm. The shaded region is bounded by the chord $AB$ and the arc $AB$ . The size of angle $AOB$ is $\frac{2}{3}\pi$ radians.

(a) Show that the area of the shaded region is approximately $0.614r^2\text{ cm}^2$ .

[2]

It is given that the radius of the circle is increasing at a rate of $0.4\text{ cm s}^{-1}$ .

(b) (i) Find the rate of increase of the area of the shaded region at the instant when $r=20$ .

Give your answer correct to 2 significant figures.

[3]

(ii) Find the rate of increase of the length of the arc $AB$ . Give your answer correct to 2 significant figures.

[3]

题目中文翻译

图中显示一个圆心为 $O$ 、半径为 $r$ cm 的扇形。阴影区域由弦 $AB$ 和弧 $AB$ 围成。角 $AOB$ 的大小为 $\frac{2}{3}\pi$ 弧度。

(a) 证明阴影区域的面积约为 $0.614r^2\text{ cm}^2$ 。

已知该圆的半径正以 $0.4\text{ cm s}^{-1}$ 的速率增加。

(b) (i) 求当 $r=20$ 时，阴影区域面积的增加速率。答案精确到 2 位有效数字。

(ii) 求弧 $AB$ 长度的增加速率。答案精确到 2 位有效数字。

CIE 9709 2025 Nov Paper 11 Q7

题目

解答