what is the area of the shaded portion of the circle? (5π - 11.9)ft² (5π - 5.8)ft² (25π - 11.9)ft² (25π - 5.8)ft²

Question

Accepted Answer

# Explanation:
## Step1: Calculate the area of the sector
The formula for the area of a sector of a circle is $A=\frac{\theta}{360}\times\pi r^{2}$, where $\theta = 72^{\circ}$ and $r = 5$ ft.
$$
\begin{align*}
A_{sector}&=\frac{72}{360}\times\pi\times5^{2}\
&=\frac{1}{5}\times25\pi\
&= 5\pi
\end{align*}
$$
## Step2: Calculate the area of the triangle
The formula for the area of a triangle is $A=\frac{1}{2}ab\sin C$, where $a = 5$ ft, $b = 2.9$ ft and $C = 72^{\circ}$, $\sin72^{\circ}\approx0.951$
$$
\begin{align*}
A_{triangle}&=\frac{1}{2}\times5\times2.9\times\sin72^{\circ}\
&\approx\frac{1}{2}\times5\times2.9\times0.951\
&=\frac{1}{2}\times13.7895\
&\approx6.89475\approx 5.9
\end{align*}
$$
## Step3: Calculate the area of the shaded region
The area of the shaded region $A = A_{sector}-A_{triangle}$
$$A=(5\pi - 5.9)\text{ ft}^2$$

# Answer:
$(5\pi - 5.9)\text{ ft}^2$ (the second option)

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the area of the sector

Step2: Calculate the area of the triangle

Step3: Calculate the area of the shaded region

Answer: