10. \begin{figure}[h] \end{figure} The triangle \(X Y Z\) in Figure 4 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 4, is a major arc of the circle with centre \(X\) and radius 4 cm .

1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region, shown shaded in Figure 4, is to be used as a design for a logo. \section*{Calculate}
3. the area of the logo
4. the perimeter of the logo.
   \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-23_383_705_1813_182} Cosine Rule: $$\begin{aligned} & \cos A = \frac { b ^ { 2 } + c ^ { 2 } - a ^ { 2 } } { 2 b c }
   & \cos A = \frac { 4 ^ { 2 } + 6 ^ { 2 } - 9 ^ { 2 } } { 2 ( 4 ) ( 6 ) }
   & A = 2.22 \text { radians } \end{aligned}$$ b)
   \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-24_323_429_302_353} $$\begin{aligned} \text { Area } & = 1 / 2 r ^ { 2 } \theta
   & = 1 / 2 ( 4 ) ^ { 2 } ( 2 \pi - 2.22 )
   & = 32.5 \mathrm {~cm} ^ { 2 } \end{aligned}$$ c) Area (Logo) \(=\) Area ( \(\Delta\) ) + Area (Sector)
   \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-24_193_268_840_620}
   \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-24_186_390_934_1028} $$\begin{aligned} & = 1 / 2 ( 4 ) ( 6 ) \sin 2.22
   \text { Area } ( \log 0 ) & = 1 / 2 ( 4 ) ( 6 ) \sin 2.22 + 1 / 2 ( 4 ) ^ { 2 } ( 2 \pi - 2.22 )
   & = 42.1 \mathrm {~cm} ^ { 2 } \end{aligned}$$ d) \(P = 9 + 2 +\) Arc length $$\begin{aligned} P & = 9 + 2 + 4 ( 2 \pi - 2.22 )
   & = 27.3 \mathrm {~cm} \end{aligned}$$ \section*{PMT PhysicsAndMathsTutor.com}

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