Long Questions (Question 9)

Question 9 (7 marks):
Mathematics Society of SMK Mulia organized a competition to design a logo for the society.


Diagram shows the circular logo designed by Adrian. The three blue coloured regions are congruent. It is given that the perimeter of the blue coloured region is 20π cm.
[Use π = 3.142]
Find
(a) the radius, in cm, of the logo to the nearest integer,
(b) the area, in cm², of the yellow coloured region.

Solution:
(a)
$\begin{array}{l}6\text{ arcs }=20\pi \\ 6r\theta =20\pi \\ 6r\left[\cancel{{60}^o}\times \frac{\pi }{{\cancel{{180}^o}}^3}\right]=20\pi \\ 2\pi r=20\pi \\ r=10\text{ cm}\end{array}$

(b)

$\begin{array}{l}\text{Area of yellow coloured region}\\ =3\left[\text{area of triangle }OAB\right]-6\left[\text{area of segment}\right]\\ =3\left[\frac{1}{2}ab\sin C\right]-6\left[\frac{1}{2}{r}^2\left(\theta -\sin \theta \right)\right]\\ =3\left[\frac{1}{2}\left(10\right)\left(10\right)\sin {120}^o\right]-6\left[\frac{1}{2}{\left(10\right)}^2\left(\theta -\sin \theta \right)\right]\\ =3\left(43.3013\right)-6\left[50\left(1.0473-\sin 1.0473\right)\right]\leftarrow \boxed{\begin{array}{l}\text{change to rad mode}\\ \theta ={60}^o\times \frac{3.142}{{180}^o}=1.0473\end{array}}\\ =129.9039-6\left(9.0612\right)\\ =129.9039-54.3672\\ =75.54{\text{ cm}}^2\end{array}$