In the figure given below, a goat is tied to a pole (at point O) which is the center of the semi-circular plot. The area of the plot is equal to its perimeter. BO is the length of the rope through which the goat is tied to the pole. Answer to the closest decimal.

The area of the plot is

The correct option is 21.83.

Step-by-step Explanation:

Let $r$ be the radius of the semi-circular plot with center $O$.

From the given image, the semi-circular region is bounded by the straight diameter $AC$ (length equal to $2r$) and the curved arc $ABC$. The dashed line $BO$ represents the radius perpendicular to the diameter.

1. Formulae for Area and Perimeter of a Semi-circle:

$\text{Area of the plot}=\frac{1}{2}\pi r^2$

$\text{Perimeter of the plot}=\pi r+2r$

2. Equating Area and Perimeter:

According to the problem, the area of the semi-circular plot is equal to its perimeter:

$\frac{1}{2}\pi r^2=\pi r+2r$

Since the radius $r>0$, we can divide both sides by $r$:

$\frac{1}{2}\pi r=\pi +2$

Solving for $r$:

$r=\frac{2(\pi +2)}{\pi }=2+\frac{4}{\pi }$

Substituting $\pi \approx 3.14159$:

$r\approx 2+\frac{4}{3.14159}\approx 2+1.27324=3.27324$

3. Calculating the Area of the Plot:

$\text{Area}=\frac{1}{2}\pi r^2$

$\text{Area}\approx 0.5\times 3.14159\times {\left(3.27324\right)}^2$

$\text{Area}\approx 1.5708\times 10.7141\approx 21.83$

Thus, the area of the semi-circular plot rounded to the closest decimal is 21.83.