Answer
(a) $15.28m^2$
(b) $24.33m$
Work Step by Step
(a) As shown in the figure given in the problem, the radius of the circle is $r=10m$, and angle is $\theta=72^\circ=\frac{2\pi}{5} rad$. The area of the fan $A1$ with angle $\theta$ is $A1=\frac{1}{2}r^2\theta=\frac{1}{2}\times10^2\times\frac{2\pi}{5}=20\pi\approx62.83m^2 $, and the area $A2$ of the isosceles triangle inside this fan is given by $A2=\frac{1}{2}\times r\times (r\times sin\theta)$ where the item $( r\times sin\theta)$ is the height. With $A2=\frac{1}{2}\times10\times10\times sin\frac{2\pi}{5} \approx47.55m^2$, the area of the shaded region can be found as $A1-A2=15.28m^2$
(b) The length $L1$ of the arc can be found as $L1=r\theta=10\times \frac{2\pi}{5} =4\pi\approx12.57m$
The unknown side $L2$ of the isosceles triangle is $L2=2\times(r\cdot sin\frac{\theta}{2})=2\times10\times sin36^\circ\approx11.76m$. Thus the perimeter of the shaded region is $L1+L2=24.33m$
