Question

The length of a rectangle is twice the diameter of a circle. The circumference of the circle is equal to the area of a square of side 22cm. What is the breadth of the rectangle, if its perimeter is 668cm?

Answer 1

Hint: The first thing we should do is to find the radius of the circle using the point that the circumference of the circle is equal to the area of the square of side 22cm. We know that the circumference of the circle is given by $2\pi r$ and the area of the square is given by ${{a}^{2}}$ , where a is its side length. Once you get the radius, double it to get the diameter and again multiply the diameter by 2 to get the length of the rectangle. Now use the formula of the perimeter of the rectangle, i.e., 2(l+b) and equate it with 668 to get the answer.

Complete step-by-step answer:
We will start the solution to the above question by finding the radius of the circle. It is given that the circumference of the circle is equal to the area of a square of side 22cm and we know that the circumference of the circle is given by $2\pi r$ and the area of the square is given by ${{a}^{2}}$ , where a is its side length.

$\text{Area of square}=\text{Circumference of the circle}$
$\Rightarrow {{a}^{2}}=2\pi r$
$\Rightarrow {{22}^{2}}=2\pi \times r$
$\Rightarrow r=\dfrac{{{22}^{2}}}{2\pi }cm$
Now, we know that the diameter of the circle is twice it’s radius and it is given that the length of the rectangle is 2 times its diameter. So, length of the rectangle is $2\times 2r=2\times 2\times \dfrac{{{22}^{2}}}{2\pi }=\dfrac{{{22}^{2}}\times 2}{\pi }cm$ .
 

Now we know that the perimeter of a rectangle is equal to 2(l+b) and also it is given in the question that its perimeter is 668cm.
$\therefore 2\left( l+b \right)=668$
We know that $l=2\times 2r=\dfrac{2\times {{22}^{2}}}{\pi }cm$ .
$\therefore 2\left( \dfrac{2\times {{22}^{2}}}{\pi }+b \right)=668$
$\Rightarrow \dfrac{2\times {{22}^{2}}}{\pi }+b=334$
Now we will put $\pi =\dfrac{22}{7}$ . On doing so, we get
$\dfrac{2\times {{22}^{2}}\times 7}{22}+b=334$
$\Rightarrow 308+b=334$
$\Rightarrow b=26cm$
Therefore, the breadth of the rectangle is equal to 26cm.

Note:The key to the above question is remembering the formulas of the perimeter and areas related to square, rectangle and circle. If you know the formulas then make sure that you don’t make a calculation error, as the only place where a mistake can be made is the calculation part. Also, taking the right value of $\pi $ out of the decimal and fraction form can ease some of your calculation part.