Composite Figures, SM-Bank 033

The design below is made up of one sector with an angle of \(\theta^\circ\) and one equilateral triangle.

Calculate the the value of \(\large \theta\) if the perimeter of the shape in terms of \(\large \pi\) is \((27+3\pi)\) metres. (3 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(60^\circ\)

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\(\text{Radius of sector}\ r=9\ \text{m, Sector angle}=\theta^\circ\)

\(\text{Perimeter}=27+3\pi\)

\(\text{Perimeter}\)	\(=3\ \text{straight edges}+1\ \text{arc length}\)
\(27+3\pi\)	\(=3\times 9+\Bigg(\dfrac{\theta}{360}\times 2\pi \times 9\Bigg)\)
\(27+3\pi\)	\(=27+\Bigg(\dfrac{\theta \times 18\pi}{360}\Bigg)\)
\(\dfrac{\theta\times \pi}{20}\)	\(=3\pi\)
\(\theta\)	\(=3\times 20=60^\circ\)

\(\therefore\ \theta=60^\circ\)