Question

The area of a circular field is equal to the area of a rectangular field. The ratio of the length and the breadth of the rectangle field is $ 14:11 $ respectively and perimeter is $ 100meters $ . What is the diameter of the circular field?
A. $ 14\;m $
B. $ 22\;m $
C. $ 24\;m $
D. $ 28\;m $

Answer 1

Hint: Here, we are given the ratio of length and breadth and the perimeter of the rectangular field. By using these two, we will find the length and breadth and then area of the rectangular field. Then, we will equate this area to the area of the circular field to find the diameter of the circular field.
Formulas used:
Perimeter of a rectangle \[ = 2\left( {l + b} \right)\] , where, $ l $ is the length of the rectangle and $ b $ is the breadth of the rectangle.
Area of a rectangle \[ = l \times b\] , where, $ l $ is the length of the rectangle and $ b $ is the breadth of the rectangle.
Area of a circle \[ = \dfrac{\pi }{4}{d^2}\] , where \[d\] is the diameter of a circle.

Complete step-by-step answer:
We are given that the ratio of the length and the breadth of the rectangular field is $ 14:11 $
$
  l:b = 14:11 \\
   \Rightarrow \dfrac{l}{b} = \dfrac{{14}}{{11}} \\
   \Rightarrow l = \dfrac{{14}}{{11}}b \;
  $
The perimeter of the rectangular field is $ 100\;m $ .
We know that Perimeter of a rectangle \[ = 2\left( {l + b} \right)\]
 $
   \Rightarrow 2(l + b) = 100 \\
   \Rightarrow (l + b) = 50 \;
  $
We will put $ l = \dfrac{{14}}{{11}}b $ in this equation
 $
   \Rightarrow \dfrac{{14}}{{11}}b + b = 50 \\
   \Rightarrow \dfrac{{25}}{{11}}b = 50 \\
   \Rightarrow b = 50 \times \dfrac{{11}}{{25}} \\
   \Rightarrow b = 2 \times 11 \\
   \Rightarrow b = 22\;m \;
  $
We know that
 $
  l = \dfrac{{14}}{{11}}b \\
   \Rightarrow l = \dfrac{{14}}{{11}} \times 22 \\
   \Rightarrow l = 14 \times 2 \\
   \Rightarrow l = 28\;m \;
  $
Now we will find the area of the rectangular field.
Area of the rectangular field $ = l \times b = 28 \times 22 = 616\;\;{m^2} $
Now, we are given that the area of the circular field is the same as the area of the rectangular field.
Area of the circular field \[ = \dfrac{\pi }{4}{d^2}\]
 \[
   \Rightarrow \dfrac{\pi }{4}{d^2} = 616 \\
   \Rightarrow {d^2} = 616 \times \dfrac{4}{\pi } \\
   \Rightarrow {d^2} = 616 \times 4 \times \dfrac{7}{{22}} \\
   \Rightarrow {d^2} = 28 \times 4 \times 7 \\
   \Rightarrow {d^2} = 28 \times 28 \\
   \Rightarrow d = 28\;m \;
 \]
Thus, the diameter of the circular field is $ 28\;m $ .
So, the correct answer is “Option D”.

Note: Here, we have used the given relations between length and breadth of the rectangular field and perimeter of it to form the linear equations. Thus, we have two linear equations and two variables, by solving which we have determined the length and breadth of the rectangular field. After that, we have equated this area with the area of the circular field and got the required answer.