题目
Problem
The diagram shows a symmetrical plate A B C D E F ABCDEF A B C D E F A B C D ABCD A B C D B C BC B C 2 2 2 A B F ABF A B F D C E DCE D C E r r r A B F ABF A B F D C E DCE D C E 1 3 π \frac{1}{3}\pi 3 1 π
(a) It is given that r = 0.4 r = 0.4 r = 0.4
(i) Show that the length E F = 2.4 EF = 2.4 E F = 2.4
[2]
(ii) Find the area of the plate. Give your answer correct to 3 3 3
[4]
(b) It is given instead that the perimeter of the plate is 6 6 6
Find the value of r r r 3 3 3
[4]
题目中文翻译
图中显示一个对称薄板 A B C D E F ABCDEF A B C D E F A B C D ABCD A B C D B C BC B C 2 2 2 A B F ABF A B F D C E DCE D C E r r r A B F ABF A B F D C E DCE D C E 1 3 π \frac{1}{3}\pi 3 1 π
(a) 已知 r = 0.4 r = 0.4 r = 0.4
(i) 证明长度 E F = 2.4 EF = 2.4 E F = 2.4
[2]
(ii) 求薄板的面积。答案保留 3 3 3
[4]
(b) 改为已知薄板的周长为 6 6 6
求 r r r 3 3 3
[4]
解答
(a)(i)
解法一
思路
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底边 E F EF E F B C BC B C
答题过程
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Each horizontal extension is 0.4 cos ( π / 3 ) = 0.2 0.4\cos(\pi/3)=0.2 0.4 cos ( π /3 ) = 0.2 E F = 2 + 2 ( 0.2 ) = 2.4 EF=2+2(0.2)=2.4 E F = 2 + 2 ( 0.2 ) = 2.4
(a)(ii)
解法一
思路
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金属板由中间的梯形和两端的扇形组成。梯形的高是半径的竖直投影。
答题过程
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The trapezium height is 0.4 sin ( π / 3 ) = 3 / 5 0.4\sin(\pi/3)=\sqrt3/5 0.4 sin ( π /3 ) = 3 /5 2 2 2 2.4 2.4 2.4
2 + 2.4 2 ⋅ 3 5 + 2 ( 1 2 ( 0.4 ) 2 π 3 ) = 11 3 25 + 4 π 75 ≈ 0.930 cm 2 \begin{align*}
&\,\frac{2+2.4}{2}\cdot\frac{\sqrt3}{5}
+2\Big(\frac12(0.4)^2\frac\pi3\Big)\\
=&\,\frac{11\sqrt3}{25}+\frac{4\pi}{75}\\
\approx&\,0.930\text{ cm}^2
\end{align*} = ≈ 2 2 + 2.4 ⋅ 5 3 + 2 ( 2 1 ( 0.4 ) 2 3 π ) 25 11 3 + 75 4 π 0.930 cm 2 to three significant figures.
(b)
解法一
思路
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沿外边界相加：上边 A D AD A D E F EF E F
答题过程
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The top edge is 2 + 2 r 2+2r 2 + 2 r 2 + 2 r cos ( π / 3 ) = 2 + r 2+2r\cos(\pi/3)=2+r 2 + 2 r cos ( π /3 ) = 2 + r r π / 3 r\pi/3 r π /3
4 + 3 r + 2 π r 3 = 6 ⟹ r = 6 9 + 2 π ≈ 0.393 cm \begin{align*}
4+3r+\frac{2\pi r}{3}=&\,6\\
\quad\Longrightarrow\quad
r=&\,\frac6{9+2\pi}\approx0.393\text{ cm}
\end{align*} 4 + 3 r + 3 2 π r = ⟹ r = 6 9 + 2 π 6 ≈ 0.393 cm to three significant figures.
