Question

(1) 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 cm2. Find the length of the wire.

Answer 1

A rectangle having a length 2x2x and breadth xxhas a perimeter of 6x6x and an area of 2x22x2.

A circle having a circumference 6x6x has a radius of 6x2π=3xπ6x2π=3xπ and an area of π(3xπ)2=9πx2π(3xπ)2=9πx2

Since the area of the circle is greater than that of the square by 104.5104.5

9πx2=2x2+104.59πx2=2x2+104.5

9πx2−2x2=104.59πx2−2x2=104.5

x2=104.59π−2=120.8387x2=104.59π−2=120.8387

∴x=10.99∴x=10.99

The length of the rectangle is 2×10.99=21.98 cm2×10.99=21.98 cm

and the length of the wire is 6×10.99=65.94 cm

Answer 2

Answer:

66 cm

Step-by-step explanation:

(i)

Let the breadth of the rectangle be 'x'.

Then the length of the rectangle = 2x.

Area of rectangle = Length * breadth

                             = 2x²

Perimeter of rectangle = 2(length + breadth)

                                     = 2(x + 2x)

                                     = 6x.

(ii)

Circumference of the circle = 2πr

⇒ 2πr = 6x

⇒ r = (3x)/π

Area of circle = πr²

                      = π(3x/π)²

                      = 9x²/π

Given that Area of circle is greater than that of rectangle by 105.4 cm².

⇒ 9x²/π = 104.5 + 2x²

⇒ 9x²/π - 2x² = 104.5

⇒ (63x²)/22 - 2x² = 104.5

⇒ 63x² - 44x² = 2299

⇒ x² = 121

⇒ x = 11 cm.

∴ Length of the rectangle = 2(22) = 44 cm.

∴ Length of the wire = 6x = 6(11) = 66 cm.

Hope it helps!