Question

The circumference of a circular ground is 88 metres. A strip of land, 3 metres wide, inside and along the circumference of the ground is to be levelled. What is the budgeted expenditure if the levelling costs Rs. 7 per square metre?
A. 1050
B. 1125
C. 1325
D. 1650

Answer 1

Hint: We will first find the radius of the circular ground from the given circumference of the ground. Next, find the inner radius. Then, the area of the circular path can be calculated by subtracting the area of the inner circle from the outer circle. Next, multiply the area of the path with the cost of levelling per square metre.

Complete step by step Answer:

We are given the circumference of the circular ground as 88 m.
We also know that the circumference of a circle is given as $C = 2\pi r$, where $r$ is the radius of the circle.
Substitute the value of circumference as 88m and $\pi = \dfrac{{22}}{7}$
Hence,
$
  88 = 2\left( {\dfrac{{22}}{7}} \right)r \\
   \Rightarrow r = \dfrac{{88\left( 7 \right)}}{{44}} \\
   \Rightarrow r = 14m \\
 $
Now, there is a strip inside the circular playground.
We want to calculate the area of the circular path.
Since the circular path is 3 metres wide, the radius of the inner circle will be $14 - 3 = 11m$

Thus, to calculate the area of the circular strip, subtract the area of the inner circle from the area of the outer circle.
We know that the area of the circle is given by $\pi {r^2}$, where $r$ is the radius of the circle.
Hence, the area of the outer circle with 14cm radius is $\pi {\left( {14} \right)^2}$
And the area of the inner circle with radius 11cm is $\pi {\left( {11} \right)^2}$
Thus, area of the circular strip is $\pi {\left( {14} \right)^2} - \pi {\left( {11} \right)^2}$
On solving the above expression, we get,
$
  \pi \left( {{{\left( {14} \right)}^2} - {{\left( {11} \right)}^2}} \right) = \dfrac{{22}}{7}\left( {14 + 11} \right)\left( {14 - 11} \right) \\
   \Rightarrow 235.72{m^2} \\
$
Now, we are given that the cost of levelling cost is Rs. 7 per square metre.
We want to find the cost to level the circular path.
Hence multiply the area of the circular path by 7 to find the total cost.
Therefore the total cost is $235.72\left( 7 \right) = Rs.1650$
Hence, option D is correct.

Note: Since, it is given that the strip of land is inside the park, then the given circumference is of the outer circle. Some students make mistakes by adding the radius calculated from circumference to the width of the path, which is incorrect. The area is always measured in square units.