Question

A wire is bent in the form of a rectangle having length twice the breadth. The same wire is bent in the form of a circle. It was found that the area of the circle is greater than that of the rectangle by $104.5{cm}^2.$ Find the length of the wire.

Answer 1

Given,

A wire is bent in shape of rectangle with length a & breadth b.

According to question

$[a=2b]$

Area of rectangle $=l\times b=a\times \frac{a}{2}=\frac{a^2}{2}$ ………..$\left(1\right)$

The same wire is bent in shape of circle.

Area of circle $=\pi r^2$

where $r=\frac{2(a+b)}{2\pi }=\frac{3a}{2\pi }$

[Perimeter of rectangle $=$ Perimeter of circle]

Area of circle $=\pi \times \frac{9a^2}{4{\pi }^2}=\frac{9a^2}{4\pi }$ ………..$\left(2\right)$

Given area of circle $=$ area of rectangle $+104.5$

$\frac{9a^2}{4\pi }=\frac{a^2}{2}+104.5$

$a^2[\frac{9}{4\pi }-\frac{1}{2}]=104.5$

$a^2\left[0.21\right]=104.5$

$[a=22.30cm]b=\frac{22.30}{2}=11.15cm$

Hence length of wire $=2(a+b)$

$=2(22.30+11.15)$

$=2\left(33.45\right)$

Length $=99.90\approx 100$

Hence wire is $100$ cm long.