Question

38 The length of a rectangular garden is 12 m more than its breadth. The numerical
of its area is equal to 4 times the numerical value of its perimeter. Find the dimensions of the garden
​

Answer 1

DIAGRAM:

$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\linethickness{0.4mm}\put(7.7,3){\large\sf{A}}\put(11.2,2){\bf{x m}}\put(7.7,1){\large\sf{D}}\put(9,3.1){\bf{x - 12\ m}}\put(11.1,1){\large\sf{C}}\put(11.1,3){\large\sf{B}}\put(8,1){\line(1,0){3}}\put(8,1){\line(0,2){2}}\put(11,1){\line(0,3){2}}\put(8,3){\line(3,0){3}}\put(7.4,2){\bf{x m}}\put(9,0.8){\bf{x - 12\ m}}\end{picture}$

ANSWER:

• LENGTH = 24 m
• BREADTH = 12 m

GIVEN:

• The length of a rectangular garden is 12 m more than its breadth.

• Area is 4 times of its perimeter.

TO FIND:

• The dimensions of the garden.

EXPLANATION:

Let the length be x and the breadth be x - 12.

Area of rectangle (A) = l × b

A = x(x - 12)

A = x² - 12x

Perimeter of rectangle (P) = 2( l + b)

P = 2(x + x - 12)

P = 2(2x - 12)

P = 4x - 24

Area = 4 × Perimeter

x² - 12x = 4(4x - 24)

x² - 12x = 16x - 96

x² - 28x + 96 = 0

By spliting the middle term.

x² - 24x - 4x + 96 = 0

x(x - 24) - 4(x - 24) = 0

Take (x - 24) as common.

(x - 4)(x - 24) = 0

x = 4 and x = 24

Length = 24 m [ Breadth cannot be negative ]

Breadth = 24 - 12 = 12 m

HENCE LENGTH = 24 m AND BREADTH = 12 m.

VERIFICATION:

AREA = 4(PERIMETER)

l × b = 4(2(l + b))

24 × 12 = 4(2(12 + 24))

288 = 8(36)

288 = 288

Hence verified.

Answer 2

Answer:

Breadth = 12 m

Length= 24 m.

Step-by-step explanation:

Let the length of the garden be x.

Let breadth be y.

Therefore, according to the question,

$x=12+y$

Area = Length times breadth

= $xy$

But we know that $x=12+y$

Therefore, area= $(12+y)y$

Perimeter= 2( length + breadth)

= $2(x+y)$

$=2(12+2y)$            (substitute the value of x = 12+y)

According to the question, area= 4(perimeter) (numerically)

Therefore, $y(y+12)=4*2(12+2y)$

Opening brackets and simplifying.

$y^2+12y=96+16y$

$y^2+12y-16y-96=0$

$y^2-4y-96=0$

$y^2-12y+8y-96=0$

$y(y-12)+8(y-12)=0$

$(y+8)(y-12)=0$

Therefore, the roots are $y=-8$ or $y=12$

But, as distance cannot be negative, y=-8 is not considered.

Hence, the correct value of y is 12.

Therefore,  y = Breadth = 12 m

x = Length = 12+breadth

Length= 12+12=24 m.

Hope it helps.