Question

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

Answer 1

Hint: Here we first need to find the width of the given rectangle in terms of the given variable. We will use the formula of the perimeter of a rectangle here; from there we will get the width in terms of the required variable. Then we will use the formula of area of rectangle to find the area of the rectangle board using the dimensions which are in terms of variables and we will put the given value of area of rectangular board in the formula. From there, we will get the value of length and width of the rectangular board.

Complete step-by-step answer:
It is given that the perimeter of the rectangular board is 70 cm and length of the rectangular board is \[x\] cm.
We know that the perimeter of rectangular is given by
$Perimeter = 2\left( {length + width} \right)$
Now, we will substitute the value of the perimeter of the rectangular board and the given length of the rectangular board.
$ \Rightarrow 70 = 2\left( {x + width} \right)$
Dividing both sides by 2, we get
$ \Rightarrow 35 = x + width$
On further simplification, we get
$ \Rightarrow width = \left( {35 - x} \right)cm$ ……… $\left( 1 \right)$
Thus, the width of the rectangular board is $\left( {35 - x} \right)cm$.
It is given that the area of the rectangular board is equal to \[300c{m^2}\].
We know that the area of rectangle is
$Area = length \times width$
Now, we will substitute the value of area, length and width of the rectangular board in the formula.
 $ \Rightarrow 300 = x \times \left( {35 - x} \right)$
On multiplying the terms, we get
$ \Rightarrow 300 = 35x - {x^2}$
On rearranging the terms, we get
$ \Rightarrow {x^2} - 35x + 300 = 0$
Now, we will factorize the obtained quadratic equation.
$ \Rightarrow {x^2} - 20x - 15x + 300 = 0$
On further simplification, we get
$
   \Rightarrow x\left( {x - 20} \right) - 15\left( {x - 20} \right) = 0 \\
   \Rightarrow \left( {x - 20} \right)\left( {x - 15} \right) = 0 \\
$
Therefore, we get
$x = 20$ or $x = 15$
We know from equation that $width = \left( {35 - x} \right)cm$
When $x = 20$
 $width = \left( {35 - 20} \right)cm = 15cm$
When $x = 15$
 $width = \left( {35 - 15} \right)cm = 20cm$
Thus, the dimensions of the rectangular board are 20 cm and 15cm.
Hence, the correct option is option A.

Note: Since we have obtained the perimeter and the area of the rectangular board here. Perimeter of a rectangle is defined as the sum of all sides of a rectangle and the area of a rectangle is defined as the product or multiplication of length and breadth of a rectangle.