Question

The area of a circular playground is 22176 sq m. The cost of fencing this ground at the rate of Rs 50 per metre is(a) Rs 52800(b) Rs 26400(c) Rs 13200(d) Rs 10560

Hint: First of all, we will draw a figure to understand the question better. It is to be kept in mind that fencing is done on the outer boundaries of the playground. The length of the outer boundary is also known as the perimeter of that shape. Thus, we need to find the perimeter or circumference of the circular ground. To get the circumference, we need the radius. We can find the radius of the circular ground based on the given area. Area of a circle is given by the relation $\pi {{r}^{2}}$ , where r is the radius. Once, we are able to find the radius, we can find the circumference given by the relation $2\pi r$ . So, circumference is the length which needs to be fenced. We are given a price for one meter of fencing. We need to multiply this cost with the circumference to find the total cost of fencing.

The figure of the circular ground is as follows.

The area of the circular ground is given as 22176 sq m.
We know that the area of the circle is given by the relation $A=\pi {{r}^{2}}$ , where A is the area and r is the radius of the circle.
We will substitute A = 22176 in the equation $A=\pi {{r}^{2}}$ and find the value of r from the equation.
\begin{align} & \Rightarrow 22176=\dfrac{22}{7}{{r}^{2}} \\ & \Rightarrow {{r}^{2}}=1008\times 7 \\ & \Rightarrow {{r}^{2}}=7056 \\ & \Rightarrow r=84 \\ \end{align} Therefore, the radius of the circular ground is 84 m.
Now, we will find the circumference of the circular ground, which is given by the relation $C=2\pi r$ , where C is the circumference of the circle.
We will substitute r = 84 in the equation $C=2\pi r$ and find the circumference.
\begin{align} & \Rightarrow C=2\times \dfrac{22}{7}\times 84 \\ & \Rightarrow C=44\times 12 \\ & \Rightarrow C=528 \\ \end{align}
Therefore, the circumference of the circle is 528 m.
We know that the cost for fencing one meter is Rs 50.
So, cost of fencing 528 m will be $528\times 50$ = Rs 26400.
So, the correct answer is “Option B”.

Note: Students are advised to be careful while finding the radius and the circumference, as the figures are very large. We’ve taken $\pi =\dfrac{22}{7}$ instead of 3.14. The reason for this is that 22 completely divides 22176. If that wasn’t the case, we shall take $\pi =3.14$ .