Question

The length of a rectangle is (4x+7)cm and the width is (5x-4)cm.
The area of the rectangle is 209cm^2
Find the perimeter of this rectangle

Answer 1

Given :

Length of the rectangle = (4x + 7) cm

Breadth of the rectangle = (5x - 4) cm

Area of the rectangle = 209 cm²

To find :

Perimeter of the rectangle

Concept :

Here, firstly we will find the value of x and substitute it in the dimensions of the rectangle to find the perimeter. To find the value of x we will use the formula of area of rectangle.

Formula of area of rectangle :-

• Area of rectangle = l × b

Formula of perimeter of rectangle :-

• Perimeter of rectangle = 2(l + b)

where,

l denotes the length of the rectangle

b denotes the breadth of the rectangle

Solution :

⟶ Area of rectangle = l × b

⟶ Substituting the given values :-

⟶ 209 = (4x + 7)(5x - 4)

⟶ 209 = 4x(5x - 4) + 7(5x - 4)

⟶ 209 = 20x² - 16x + 35x - 28

⟶ 209 = 20x² + 19x - 28

⟶ 20x² + 19x - 28 - 209 = 0

⟶ 20x² + 19x - 237 = 0

⟶ A quadratic equation is formed and it cannot be solved further. So, here we will use the quadratic formula.

Quadratic formula :-

$\boxed{\bold{x = \dfrac{ -b \: \pm \: \sqrt{ {b}^{2} - 4ac} }{2a} } }$

we have,

• a = 20
• b = 19
• c = - 237

Substituting the given values,

$\\ \longrightarrow \quad \sf{x = \dfrac{ -b \: \pm \: \sqrt{ {b}^{2} - 4ac} }{2a} }$

$\\ \longrightarrow \quad \sf{x = \dfrac{ -19 \: \pm \: \sqrt{ {19}^{2} - 4(20)( - 237)} }{2(20)} }$

$\\ \longrightarrow \quad \sf{x = \dfrac{ -19 \: \pm \: \sqrt{ 361 + 18960} }{40} }$

$\\ \longrightarrow \quad \sf{x = \dfrac{ -19 \: \pm \: \sqrt{19321} }{40} }$

$\\ \longrightarrow \quad \sf{x = \dfrac{ -19 \: \pm \: 139 }{40} }$

$\\ \longrightarrow \quad \sf{x = \dfrac{ -19 \: + \: 139 }{40} } \quad or \quad \sf{x = \dfrac{ -19 \: - \: 139 }{40} }$

$\\ \longrightarrow \quad \sf{x = \dfrac{ 120 }{40} } \quad or \quad \sf{x = \dfrac{ -158 }{40} }$

$\\ \longrightarrow \quad \sf{x = 3 \quad or \quad \sf{x = -3.95 }}$

The dimensions of the rectangle cannot be negative. So, the negative sign will get rejected.

Therefore, the value of x = 3

Substituting the value of x :-

LENGTH :

⟶ Length of the rectangle = (4x + 7)

⟶ Length of the rectangle = (4(3) + 7)

⟶ Length of the rectangle = 12 + 7

⟶ Length of the rectangle = 19 cm

BREADTH :

⟶ Breadth of the rectangle = (5x - 4)

⟶ Breadth of the rectangle = (5(3) - 4)

⟶ Breadth of the rectangle = 15 - 4

⟶ Breadth of the rectangle = 11 cm

Perimeter of the rectangle :-

⟶ Perimeter of rectangle = 2(l + b)

⟶ Perimeter of rectangle = 2(19 + 11)

⟶ Perimeter of rectangle = 2(30)

⟶ Perimeter of rectangle = 60

Therefore,

• The perimeter of rectangle = 60 cm