(a)
Change the angle to radian,
\(75^\circ\times \dfrac{\pi}{180^\circ}=1.3092\text{ rad}.\)
Use the formula of area of sector,
\(\text{Area}=\dfrac{1}{2}r^2\theta\)
Area of shaded region:
\(=\text{Area of sector }OAB\text{Area of sector }OCD\)
\(\begin{aligned} 13.09&=\dfrac{1}{2}(3k)^2(1.3092)\dfrac{1}{2}(2k)^2(1.3092) \\\\ 13.09&=\dfrac{1}{2}(9)(1.3092)k^2\dfrac{1}{2}(4)(1.3092)k^2 \\\\ 13.09&=5.8914k^22.6184k^2 \\\\ 13.09&=3.273k^2 \\\\ k&=\sqrt{\dfrac{13.09}{3.273}}\\\\ k&=1.9998\approx2\text{ cm}. \end{aligned}\)
(b)
Use the formula of arc length:
\(s=r\theta\)
Perimeter of the shaded region:
\(=k+\text{Arc }CD+k+\text{Arc }AB\)
\(\begin{aligned} &=2+2+2(2)(1.3092)+3(2)(1.3092) \\ &=4+5.2368+7.8552 \\ &=17.092\text{ cm}. \end{aligned}\)
