$ABC$ is a sector of a circle with radius $AB$ and $AC$ and $\measuredangle BAC=\theta $. \(

Question

$ABC$ is a sector of a circle with radius $AB$ and $AC$ and $\measuredangle BAC=\theta $. $D$ is a point on $AC$ such that $AD=x$cm, $CD=y$m and $\measuredangle ADB=\frac{\pi }{2}$. Show that $BD=\sqrt{{{y}^{2}}+2xy}$.[1] Hence show that $\sin \theta \approx {{\left( 2\alpha \right)}^{\frac{1}{2}}}\left( 1-\frac{3}{4}\alpha \right)$, where $\alpha =\frac{y}{x}$, and $y$ is small compared to $x$.[4] You are now given that $\theta =10{}^\circ $ and $x=100$. Use part (ii) to find estimates for $\alpha $ and $y$ correct to $2$ decimal places. Find the actual value of $y$.[1] Comparing the values of $y$ obtained in parts (iii) and (iv), comment on the accuracy of your approximations and explain your answer.[1]

Accepted Answer

(i) $BD=\sqrt{y^2+2xy}$ (ii) $\sin\theta\approx\sqrt{2\alpha}\left(1-\frac34\alpha\right)$ (iii) $\alpha=0.02$, $y=1.54\text{ cm}$ (iv) $y=1.54\text{ cm}$ (v) The estimate is accurate to $2$ d.p. because $\alpha\approx0.0154$ is small

2021 SAJC P2 Q4

2021 SAJC P2 Q4

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