题目

Problem

Figure 1 shows a sketch of the entrance to a tunnel.

Figure 1

The shape of the entrance consists of a sector $BCDF$ , of a circle centre $F$ , joined to two congruent (identical) triangles $ABF$ and $EDF$ .

Given that

\begin{align*} AFE\text{ is a straight line} \end{align*}

\begin{align*} AF=FE=6.4\text{ m} \end{align*}

\begin{align*} FB=FD=6.2\text{ m} \end{align*}

\begin{align*} \text{angle }BFD=2.275\text{ radians} \end{align*}

(a) Show that angle $AFB=0.433$ radians to $3$ decimal places.

(1)

(b) Find the perimeter of the entrance to the tunnel, $ABCDEFA$ , in metres, to one decimal place.

(4)

解答

(a)

解法一

思路

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$AFE$ 是直线，所以围绕点 $F$ 的上半部分总角度是 $\pi$ 。中间扇形角是 $2.275$ ，两侧三角形全等，所以两侧角相等。

答题过程

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Since $AFE$ is a straight line and the two triangles are congruent,

\begin{align*} \angle AFB =&\,\frac{\pi-2.275}{2}\\ =&\,0.433296\ldots\\ =&\,0.433\text{ radians} \end{align*}

to $3$ decimal places.

(b)

解法一

思路

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周长由底边 $AF+FE$ 、两条斜边 $AB,DE$ 和弧长 $BCD$ 组成。弧长用 $r\theta$ ；斜边 $AB$ 可在三角形 $ABF$ 中用余弦定理求出。

答题过程

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The arc length $BCD$ is

\begin{align*} r\theta =&\,6.2(2.275)\\ =&\,14.105. \end{align*}

Using the cosine rule in triangle $ABF$ ,

\begin{align*} AB^2 =&\,6.4^2+6.2^2-2(6.4)(6.2)\cos(0.433296\ldots)\\ =&\,7.367\ldots. \end{align*}

\begin{align*} AB=2.7142\ldots. \end{align*}

The perimeter is

\begin{align*} 2AB+\text{arc }BCD+AF+FE =&\,2(2.7142\ldots)+14.105+12.8\\ =&\,32.333\ldots. \end{align*}

Therefore the perimeter is

\begin{align*} 32.3\text{ m}. \end{align*}

(c)

解法一

思路

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总面积等于中间扇形面积加两侧全等三角形面积。扇形面积用 $\frac12r^2\theta$ ，三角形面积用 $\frac12ab\sin C$ 。

答题过程

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The area of sector $BCDF$ is

\begin{align*} \frac12r^2\theta =&\,\frac12(6.2)^2(2.275)\\ =&\,43.7255. \end{align*}

The area of triangle $ABF$ is

\begin{align*} \frac12(6.4)(6.2)\sin(0.433296\ldots) =8.325\ldots. \end{align*}

There are two congruent triangles, so the total area is

\begin{align*} 43.7255+2(8.325\ldots) =&\,60.376\ldots. \end{align*}

Therefore the cross-sectional area is

\begin{align*} 60.4\text{ m}^2. \end{align*}

IAL 2024 May R Q6

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程