Question

Length of a rectangle is 5 cm longer than the width. Four squares are constructed outside the rectangle such that each of the squares share one side with the rectangle. The total area of the constructed figure is 120 cm2. What is the perimeter of the rectangle?

Answer 1

Answer:

The perimeter of rectangle is $18\ cm$

Step-by-step explanation:

Let

x-----> the length of the rectangle

y----> the width of the rectangle

we know that

$x=y+5$ ----> equation A

$120=xy+2x^{2}+2y^{2}$  ---> equation B (area of the constructed figure)

substitute the equation A in equation B

$120=(y+5)y+2(y+5)^{2}+2y^{2}$

$120=(y+5)y+2(y+5)^{2}+2y^{2}\\ 120=y^{2}+5y+2(y^{2}+10y+25)+2y^{2}\\ 120=y^{2}+5y+2y^{2}+20y+50+2y^{2}\\120=5y^{2}+25y+50\\5y^{2}+25y-70=0$

using a graphing calculator -----> solve the quadratic equation

The solution is

$y=2\ cm$

Find the value of x

$x=y+5 ----> x=2+5=7\ cm$

Find the perimeter of rectangle

$P=2(x+y)=2(7+2)=18\ cm$

Answer 2

Answer:

The perimeter of the rectangle = 18 cm

Step-by-step explanation:

* The figure consists of one rectangle and four squares

* Lets put the dimensions of the rectangle and the four squares

- The width of the rectangle is x

∵ The length of the rectangle is 5 cm longer than the width

∴ The length of the rectangle is x + 5

∵ Area the rectangle = L × W

∴ Area rectangle = x(x + 5) = x² + 5x ⇒ (1)

* There are four squares constructed each in one side

  of the rectangle

- Two squares constructed on the length of the rectangle

∴ The length of the sides of the squares = x + 5

∵ Area the square = S²

∴ Its area = (x + 5)² ⇒ use the foil method

∴ Its area = x² + 10x + 25

∵ They are two squares

∴ Its area = 2x² + 20x + 50 ⇒ (2)

- Two squares constructed on the width of the rectangle

∴ The length of the sides of the squares = x

∴ Its area = x²

∵ They are two squares

∴ Its area = 2x² ⇒ (3)

- Area rectangle = x(x + 5) = x² + 5x ⇒ (3)

* Now lets make the equation of the area of the figure by

 adding (1) , (2) and (3)

∴ The area of the figure = 2x² + 20x + 50 + 2x² + x² + 5x  

                                         = 5x² + 25x + 50⇒ (4)

∵ The area of the figure = 120 cm² ⇒ (5)

* Put (4) = (5)

∴ 5x² + 25x + 50 = 120 ⇒subtract 120 from both sides

∴ 5x² + 25x +50 - 120 = 0 ⇒ add the like term

∴ 5x² + 25x -70 = 0 ⇒ divide both sides by 5

∴ x² + 5x - 14 = 0 ⇒ factorize it to find the value of x

∴ (x - 2)(x + 7) = 0

* Lets equate each bracket by 0

∴ (x - 2) = 0 ⇒ x = 2

∴ (x + 7) = 0 ⇒ x = -7 ⇒ rejected no -ve side length

∴ x = 2

* Now lets find the dimensions of the rectangle

∵ Length rectangle = x + 5

∴ Length rectangle = 2 + 5 = 7 cm

∴ Width rectangle = 2

∵ The perimeter of the rectangle = 2(L + W)

∴ The perimeter of the rectangle = 2(2 + 7) = 18 cm

* The perimeter of the rectangle = 18 cm