Question

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

Answer 1

Answer:

66 cm

Step-by-step explanation:

Define x:

Let the breadth be x

The length = 2x

Find the area of the rectangle:

Area = Length x Breadth

Area = 2x²

Find the perimeter of the rectangle:

Perimeter = 2(Length + breadth)

Perimeter = 2(2x + x)

Perimeter = 6x

Find the radius of the circle in term of x

Circumference = 2πr

2πr = 6x

r = 6x ÷ 2π

r = 3x/π

Find the area of the circle in term of x:

Area = πr²

Area = π(3x/π)² = 9x²/π

Solve x:

The area of the circle is greater than the rectangle by 104.5 cm²

9x²/π -  2x² = 104.5

x² (9/π - 2) = 104.5

19/22 x² = 104.5

x²  = 104.5 ÷ 19/22

x² = 121

x = √121

x = 11 cm

Find the length of the wire:

Length = 6x = 6(11) = 66 cm

Answer: The length is 66 cm

Answer 2

Answer:

Length of the wire is 65.94 cm

Step-by-step explanation:

Let the length of the wire is 'l'.

Let 'x' be the breath of the rectangle formed by bending the wire. As the length of this rectangle is two times its breath, so the length of this rectangle is '2x'. The perimeter of this rectangle is the sum of its sides and, thus, equals to the length of the wire. So,

$2(2x+x)=l\\2(3x)=l\\6x=l.....(i)$

Let 'r' be the radius of the circle formed from the wire. Its perimeter should be equal to the length of the wire. So,

$2\pi r=l.....(ii)$

Equating (i) and (ii),

$6x=2\pi r\\x=\frac{\pi}{3}r.....(iii)$

As the area of circle is greater than that of rectangle by 104.5 cm²,

$Area~of~circle - Area~of~rectangle = 104.5\\ \pi r^2 -2x \times x=104.5\\ \pi r^2 -2x^2=104.5\\\pi r^2-2 (\frac{\pi}{3}r )^2=104.5\\r^2 (\pi - 2(\frac{\pi}{3})^2)=104.5\\r=\sqrt{\frac{104.5}{(\pi - 2(\frac{\pi}{3})^2} }\\ r=10.5$

So, $l=2\pi r= 2\pi (10.5) =65.94~cm$