Maths-

General

Easy

Question

# The perimeter of a rectangular plot is 180 m and its area is 1800 sq.metre. Take the length of the plot as x m. Use the perimeter to write the value of the breadth in terms of x. Use the value of length , breadth and the area to write an equation in x. Solve the equation to calculate the length and breadth of the plot.

Hint:

### Perimeter of a rectangle = 2 (length + breadth)

Area of a rectangle = length × breadth

## The correct answer is: ∴ Length & Breadth of the given rectangle are 60 m & 30 m, respectively

### Step-by-step solution:-

As given in the sum,

Let the length of the given rectangle be x m

Length = x m ....................................................................................... (Equation i)

We know that-

Perimeter of a rectangle = 2 (length + breadth)

∴ Perimeter of a rectangle = 2 (x + breadth) ........................................... (From Equation i)

∴ 180 = 2 (x + breadth) ........................................... (From given information)

∴ 180 / 2 = x + breadth

∴ 90 = x + breadth

∴ 90 - x = breadth .................................................... (Equation ii)

Area of a rectangle = length × breadth

∴ Area of a rectangle = x × (90 - x) ........................................................ (From Equations i & ii)

∴ 1,800 = 90x - x^{2} ........................................................... (From given information)

∴ x^{2} - 90x + 1,800 = 0

Comparing this quadratic equation with the general form, i.e. ax2 + bx + c = 0, we get-

a = 1, b = -90, c = 1,800

We can solve the above quadratic equation using factorization method as follows-

Steps for Factorization method-

Step-1: a × c = 1 × 1,800 = 1,800

Step-2: Factorize 1,800 in such a way that the 2 factors, when added, gives the value b = -90

i.e. 1,800 = -30 × -60

& -30 + (-60) = -90 = b

∴ x^{2} - 90x + 1,800 = 0

∴ x^{2} - 60x - 30x + 1,800 = 0

∴ x (x - 60) - 30 (x - 60) = 0

∴ (x - 30) (x - 60) = 0

either x - 30 = 0 or x - 60 = 0

either x = 30 or x = 60

From Equations i & ii-

When Length = x = 30

Breadth = 90 - x = 90 - 30 = 60

When Length = 60

Breadth = 90 - x = 90 - 60 = 30

Since Length of a rectangle should be greater than its breadth,

Length = 60 & Breadth = 30

Final Answer:-

∴ Length & Breadth of the given rectangle are 60 m & 30 m, respectively.

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