Question

a wire is bent in the form of a rectangle with dimensions 5x cm by (4x+2)cm.wire B is bent to form another rectangle with dimensions (6x+3)cm by (3x+1)cm.given that the two rectangles are equal in area.Find the value of x and also determine which rectangle have greater perimeter

Answer 1

Answer:

x = 1.5

Second rectangle will have larger perimeter

Step-by-step explanation:

given that

a wire is bent in the form of a rectangle with dimensions 5x cm by (4x+2)cm

here,

length if the rectangle = 5x cm

breadth of the rectangle = (4x + 2) cm

given that,

this wire is again bented and form another rectangle of same area

whose length = (6x + 3) cm

breadth = (3x + 1) cm

since,

area of both rectangle are equal so,

5x(4x + 2) = (6x + 3)(3x + 1)

20x² + 10x = 18x² + 6x + 9x + 3

20x² - 18x² + 10x = 15x + 3

2x² + 10x - 15x = 3

2x² - 5x - 3 = 0

2x² + x - 6x - 3 = 0

2x(x + 1) -3(x + 1) = 0

(2x - 3)(x + 1) = 0

now,

2x - 3 = 0

2x = 3

x = 3/2

also,

x + 1 = 0

x = -1

since,

length can be negative,

so,

x = 3/2

= 1.5 cm

now,

length of the first rectangle

= 5x

= 5(1.5)

= 7.5 cm

breadth = 4x + 2

= 4(1.5) + 2

= 6 + 2

= 8 cm

perimeter = 2(length + breadth)

= 2(7.5 + 8)

2(15.5)

= 31 cm

now,

in the second rectangle

length = 6x + 3

= 6(1.5) + 3

= 9 + 3

= 12 cm

breadth = 3x + 1

= 3(1.5) + 1

= 4.5 + 1

= 5.5 cm

perimeter = 2(length + breadth)

= 2(12 + 5.5)

= 2(17.5)

= 35 cm

now,

we have,

perimeter of the first rectangle = 31 cm

perimeter of the second rectangle = 35 cm

so,

x = 1.5

Second rectangle will have larger perimeter

Answer 2

$\mathfrak{\large{\underline{\underline{\blue{Answer:-}}}}}$

X = 1.5 cm

$\mathfrak{\large{\underline{\underline{\blue{Explanation:-}}}}}$

A wire is bent in the form of rectangle, whose dimensions are :-

5x cm by (4x+2)cm

So, we know that

length = 5x cm

Breadth = 4x+2 cm

$\rule{200}{2}$

We know that it is again bent into a rectangle wire

Whose, dimensions are :-

Length = 6x+3 cm

Breadth = 3x+1 cm

$\rule{200}{2}$

We have same wire, So both have equal areas.

Area of rectangle1 = Area of rectangle 2

5x(4x+2) = (6x+3)(3x+1)

20x² + 10x = 18x² + 6x + 9x + 3

20x² - 18x² = 15x - 10 x + 3

2x² = 5x + 3

2x² - 5x + 3

______________[ Splitting the middle term]

2x² + x - 6x - 3 = 0

2x(x+1) - 3(x+1) = 0

(2x-3) (x+1) = 0

_______________

⬇⬇ ⬇⬇

2x - 3 = 0. x+1=0

2x = 3. x = -1 cm

x = 3/2

x = 1.5 cm

$\rule{200}{2}$

Length can't be negative,So we will put here positive value only.

X = 1.5 cm

___________[ Answer]

$\rule{200}{2}$

Triangle 1 :-

length = 5x

» 5(1.5)

» 7.5cm

____________

breadth = 4x + 2

» 4 (1.5) + 2

» 6 + 2

» 8 cm

___________________

Perimeter = 2 ( l + b)

» 2 ( 8 + 7.5)

» 2(15.5)

» 30 cm

$\rule{200}{2}$

Triangle 2 :-

length = 6x+3

» 6 (1.5) + 3

» 9 + 3

» 12 cm

______________

Breadth = 3x+1

» 3 (1.5) + 1

» 4.5 + 1

» 5.5 cm

________________________

Perimeter = 2( l + b)

» 2(12+5.5)

» 2(17.5)

» 35 cm

$\rule{200}{2}$

So, perimeter of Triangle 2 is greater.