题目

Problem

Figure 1 shows a sector $AOB$ of a circle with centre $O$ and radius $r$ cm.

Figure 1

The angle $AOB$ is $1.25$ radians.

Given that the area of the sector $AOB$ is $15\text{ cm}^2$

(a) find the exact value of $r$ ,

(2)

(b) find the exact length of the perimeter of the sector. Write your answer in simplest form.

(3)

解答

展开

扇形面积公式是 $\frac12r^2\theta$ 。这里角度已经是弧度，所以可以直接代入。

展开

\begin{align*} \frac12r^2(1.25)=&\,15\\ \frac58r^2=&\,15\\ r^2=&\,24. \end{align*}

Since $r>0$ ,

\begin{align*} r=\sqrt{24}=2\sqrt6. \end{align*}

展开

扇形周长等于两条半径加上弧长。弧长公式是 $r\theta$ 。

展开

The arc length is

\begin{align*} r\theta =&\,2\sqrt6(1.25)\\ =&\,\frac52\sqrt6. \end{align*}

The perimeter is

\begin{align*} 2r+r\theta =&\,2(2\sqrt6)+\frac52\sqrt6\\ =&\,4\sqrt6+\frac52\sqrt6\\ =&\,\frac{13\sqrt6}{2}. \end{align*}