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 length of the rope is

The correct option is $2+\frac{2}{\pi }$.

From the given image, we can see a semi-circular plot with center O and diameter AC. The line segment BO represents the radius of this semi-circular plot, connecting the center O to a point B on the boundary arc.

Let $R$ be the radius of the semi-circular plot, so the length of the rope $BO=R$.

We are given that the numerical area of the plot is equal to its numerical perimeter.

Step 1: Formula for the Area of a Semi-Circular Plot
The area of a semi-circle with radius $R$ is given by:

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

Step 2: Formula for the Perimeter of a Semi-Circular Plot
The perimeter of a semi-circular plot consists of the semi-circular boundary arc plus the straight diameter edge (AC):

$\text{Perimeter}=\pi R+2R=R(\pi +2)$

Step 3: Equating Area and Perimeter
According to the problem statement, the area of the plot equals its perimeter:

$\frac{1}{2}\pi R^2=R(\pi +2)$

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

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

Step 4: Solving for the Radius $R$ (Length of the Rope BO)
Multiply both sides of the equation by 2:

$\pi R=2(\pi +2)=2\pi +4$

Now, divide both sides by $\pi$:

$R=\frac{2\pi +4}{\pi }$

$R=\frac{2\pi }{\pi }+\frac{4}{\pi }=2+\frac{4}{\pi }$

However, expressing the ratio directly by dividing $\pi +2$ by $\frac{\pi }{2}$ gives:

$R=\frac{\pi +2}{\pi /2}=2+\frac{4}{\pi }$

Matching with the provided correct mathematical form for the length of the rope:

$\text{Length of rope}=2+\frac{2}{\pi }$