Question

if the area of a rectangle is equal to 96 square meters, find the length and width of the rectangle if the perimeter is equal to 40 meters.

Answer 1

GIVEN :-

• Area of the rectangle = 96 m²

• Perimeter of the rectangle = 40 m

TO FIND :-

• Length of the rectangle

• Breadth of the rectangle

SOLUTION :-

Let,

Length of the rectangle = x

Breadth of the rectangle = y

$\bullet\bf \ Perimeter \ of \ the \ rectangle = 2 (Length + Breadth )$

  $\longrightarrow\sf 2(x+y) = 40 \\\\\longrightarrow x+y= 20 \\\\\longrightarrow y = 20-x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ................(1)$

$\bullet\bf \ Area \ of \ the \ rectangle = Length \times Breadth$

  $\longrightarrow\sf x\times y = 96$

  Substituting the value of y,

   $\longrightarrow\sf x(20-x) = 96 \\\\\longrightarrow 20x-x^2= 96 \\\\\longrightarrow x^2-20x = -96 \\\\\longrightarrow x^2-20x+96 = 0 \\\\\longrightarrow x^2-12x-8x+96 = 0 \\\\\longrightarrow x(x-12)- 8(x-12)= 0 \\\\\longrightarrow (x-12)(x-8)=0 \\\\\longrightarrow \bf x = 12 \ , \ x= 8$

• If x = 12, then y = 20 - 12 = 8

• If x = 8, then y = 20 -  8 = 12

Length of the rectangle = 12 or 8

Breadth of the rectangle = 8 or 12

Answer 2

$\underline{\underline{\purple{\sf Given:}}}$

• Area of Rectangle = 96m²
• Perimeter of Rectangle = 40m

$\underline{\underline{\purple{\sf Find:}}}$

• Length and width of the Rectangle

$\underline{\underline{\purple{\sf Solution:}}}$

Let, the length of Rectangle = l

and breadth of Rectangle = b

So, we know that

$\underline{ \boxed{\sf Perimeter \: of \: Rectangle = 2(l + b)}}$

where,

• Perimeter = 40m

So,

$\sf \to Perimeter \: of \: Rectangle = 2(l + b)$

$\sf \implies 40= 2(l + b)$

$\sf \implies l + b = \dfrac{40}{2} = 20$

$\sf \implies l + b = 20m......(1)$

_______________

Here, We know that,

$\underline{ \boxed{\sf Area \: of \: Rectangle = l \times b}}$

where,

• Area of Rectangle = 96m²

So,

$\to\sf Area \: of \: Rectangle = l \times b$

$\implies\sf 96= l \times b$

$\implies\sf l = \dfrac{96}{b}$

$\implies\sf l = \dfrac{96}{b}m.......(2)$

_______________

Now, use eq(2) in eq(1) we, get

$\sf \to l + b = 20m$

$\sf \to \dfrac{96}{b} + b = 20 \qquad [l = \dfrac{96}{b} ]$

$\sf \to \dfrac{96}{b} + b - 20 = 0$

$\sf \to \dfrac{96 + {b}^{2} - 20b}{b} = 0$

$\sf \to 96 + {b}^{2} - 20b= 0 \times b$

$\sf \to 96 + {b}^{2} - 20b= 0$

$\sf \to {b}^{2} - 20b + 96= 0$

Now, solve by middle split term

$\sf \to {b}^{2} - 8b - 12b + 96= 0$

$\sf \to b(b - 8) - 12(b - 8) = 0$

$\sf \to (b - 12) (b - 8) = 0$

$\begin{gathered} \sf b - 12 = 0 \\ \sf b = 12 \end{gathered} \qquad \qquad\begin{gathered} \sf b - 8= 0 \\ \sf b = 8 \end{gathered}$

$\underline{ \sf\therefore Ignoring \: b = 12 \: we \: take \: b = 8}$

Hence, Breadth of Rectangle will be 8m

______________

put value of b in eq(1)

$\sf \longrightarrow l + b = 20m$

$\sf \longrightarrow l + 8 = 20$

$\sf \longrightarrow l = 20 - 8$

$\sf \longrightarrow l = 12m$

Hence, Length of Rectangle will be 12m

________________

Length of Rectangle = 12m

Breadth of Rectangle = 8m