Answer:
168.9 cm²
Step-by-step explanation:
To find the area of the shaded region, we can use the formula for the area of a sector:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a sector}}\\\\A=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\;\textsf{$\theta$ is the angle measured in degrees.}\end{array}}[/tex]
Angles around a point sum to 360°. Therefore, the central angle (θ) of the shaded area can be found by subtracting the central angle of the unshaded area from 360°:
[tex]\theta=360^{\circ}-65^{\circ}\\\\\theta=295^{\circ}[/tex]
Therefore, the values are:
Substitute the values into the formula and solve for A:
[tex]A=\left(\dfrac{295^{\circ}}{360^{\circ}}\right) \cdot\pi \cdot 8.1^2[/tex]
[tex]A=\dfrac{59}{72} \cdot \pi \cdot 65.61[/tex]
[tex]A=53.76375\pi[/tex]
[tex]A=168.903802...[/tex]
[tex]A=168.9\; \sf cm^2\;(nearest\;tenth)[/tex]
Therefore, the area of the shaded region is 168.9 cm², rounded to the nearest tenth.
