Question

The perimeter of a rectangular board is $70cm$. Taking its length as $xcm,$ find its width in terms of $x$.
If the area of the rectangular board is $300c{m^2}$, find its dimensions.
A) $Width = \left( {35 - x} \right)cm;20cm,15cm$
B) $Width = \left( {70 - x} \right)cm;30cm,15cm$
C) $Width = \left( {45 - x} \right)cm;20cm,10cm$
D) $Width = \left( {35 - x} \right)cm;10cm,30cm$

Answer 1

Hint: First we will find the width of the rectangular board in terms of $x$ with the help of perimeter given in the problem. Then we can proceed for the area of the rectangular board to get the value of the variable $x$.

Complete step-by-step answer:
Let $ABCD$ be the rectangular board whose perimeter is 70 cm as shown below.

AB = CD = x
We have been given the length of the rectangular board as $x$.
$\therefore \,AB = x$ ….…… Eq I
We know that the perimeter of a rectangle can be obtained from the following formula:
$Perimeter\, = \,2\left( {l + b} \right)$ ……… Eq II
Where,
l = Length of the rectangle
b = Breadth of the rectangle
Using I and II,
$70\, = \,2\left( {x + b} \right)$
$ \Rightarrow x + b = 35$
$ \Rightarrow b = 35 - x$
Thus, width of the rectangular board is $\left( {35 - x} \right)$
$ \Rightarrow BC = 35 - x$ ……… Eq III
We know that the area of a rectangle can be obtained from the following formula:
$A\, = \,lb$ ………Eq IV
Where,
l = Length of the rectangle
b = Breadth of the rectangle
A = Area of the rectangle
Here,
$A = AB \times BC$
Using equations, I, III and IV,
$300 = x\left( {35 - x} \right)$ [ $\because \,A = 300c{m^2}$ ]
$ \Rightarrow 300 = 35x - {x^2}$
$ \Rightarrow {x^2} - 35x + 300 = 0$
Using Middle term split method,
${x^2} - 20x - 15x + 300 = 0$
$ \Rightarrow x\left( {x - 20} \right) - 15\left( {x - 20} \right) = 0$
$ \Rightarrow \left( {x - 15} \right)\left( {x - 20} \right) = 0$
$x = 15$ or $x = 20$
Thus, length of the rectangular board $ = x = 20cm$
And the width of the rectangular board $ = 35 - x\,\,\, = 15cm$
Therefore, option (A) is correct.

Note: If we take $x = 15$ it gives us length as $x = 15$ and breadth as $35 - x = 20$. But for a rectangle, length > breadth.
Thus, we will take $x = 20$ because it gives us length as $x = 20$ and breadth as $35 - x = 15$.