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 the circle is greater than that of the rectangle by 104.5sq.cm find the length of the wire

Answer 1

Answer:

Length of wire = 725 cm

Step-by-step explanation:

It is given that the wire is used to make a rectangle, with a length that is double the width. Therefore, The area of the rectangle is $2w*w=2w^2$.

It is also given that the wire is used to make a circle whose area is greater than that of the rectangle by 104.5 sq.cm. Therefore, $\pi r^2=2w^2+104.5$.

Since the two shapes are created using the same exact wire, therefore, the circumference of both shapes should be equal.

Therefore, $2(2w+w)=2\pi r\\6w=2\pi r\\3w=\pi r\\r=\frac{3w}{\pi}$

Substitute in the area equation:

$\pi r^2=2w^2+104.5$.

$\pi (\frac{3w}{\pi })^2=2w^2+104.5\\\frac{9w^2}{\pi }=2w^2+104.5\\9w^2=2\pi w^2+104.5\pi \\w^2(9-2\pi )=104.5\pi \\w^2=\frac{104.5\pi }{9-2\pi } \\w=\sqrt{\frac{104.5\pi }{9-2\pi }}=120.83$.

Therefore, the length of the wire is equal to the perimeter of the rectangle = 6w = 6(120.83)=725 cm

Answer 2

Answer:

66 cm

Step-by-step explanation:

Define x:

Let the breadth be x

The length is 2x

Find the Perimeter of the rectangle:

Perimeter = 2(Length + Breadth)

Perimeter = 2(x + 2x) = 6x

Find the area of the rectangle:

Area = Length x Breadth

Area = 2x(x) = 2x²

Find the radius of the circle:

Circumference = 2πr

6x = 2πr

r = 6x ÷ 2π

r = 3x/π

Find the area of the circle:

Area = πr²

Area = π(3x/π)²

Area = 9x²/π

Area =63x²/22

Solve x:

Give that the area circle is 104.5 cm² greater than the rectangle.

63x²/22 -  2x² = 104.5

63x² - 44x² =  2299

19x² = 2299

x² = 121

x = √121

x = 11 cm

Find the perimeter of the wire:

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

Answer: The perimeter is 66 cm long.