Question

In the figure above, the circle with center O has a radius that is 4 units long. If the area of the shaded region is 147 square units, what is the value of x?​

Answer 1

first off, let's find the whole area of a circle with a radius of 4.

`\textit{Area of a Circle}\\\\ A=\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=4 \end{cases}\implies A=\pi (4)^2\implies A=16\pi \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{\small area that is \underline{not shaded} inside the circle}}{\stackrel{ \textit{whole area} }{16\pi} ~~ - ~~\stackrel{ \textit{shaded area} }{14\pi} } \implies 2\pi`

so hmmm, now let's find the angle θ inside the circle, whose area is 2π and of radius 4.

`\textit{Area of a Sector of a Circle}\\\\ A=\cfrac{\theta \pi r^2}{360} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=4\\ A=2\pi \end{cases}\implies 2\pi =\cfrac{\theta \pi (4)^2}{360}\implies 720\pi =16\pi \theta \\\\\\ \cfrac{720\pi }{16\pi }=\theta \implies 45=\theta \hspace{9em}x=180-90-\theta \implies \boxed{x=45^o}`