# Ex.4.4 Q5 Quadratic Equations Solutions - NCERT Maths Class 10

## Question

Is it possible to design a rectangular park of perimeter of \(80\,\rm{m}\) and area \(400\,\rm{m}^2\)?

If so, find its length and breadth.

## Text Solution

**What is Known?**

Perimeter of rectangular park \( = 80\,{\rm{ m}}\)

Area of rectangle \(=400\, \rm{m}^2\)

**What is Unknown?**

Checking the possibility to design a rectangular park with the given condition , If yes find its perimeter .

**Reasoning:**

Perimeter of rectangle

\( = 2(l + b) = 80 \; \ldots (1)\)

Area of rectangle \( = lb = 400 \; \ldots \left( 2 \right)\)

We will use the \(1^\rm{st}\) equation to express \(l\) in the form of \(b.\) Then, we will substitute this value of \(l\) in equation \(2.\)

**Steps:**

\[\begin{align}2(l + b) &= 80\\(l + b) &= 40\\l &= 40 - b\end{align}\]

Substituting the value of \(l = 40 - b\) in equation (2)

\[\begin{array}{*{20}{l}}

{(40 - b)(b) = 400}\\

{40b - {b^2} = 400}\\

{40b - {b^2} - 400 = 0}\\

{{b^2} - 40b + 400 = 0}\\

{}

\end{array}\]

Let’s find the discriminant: \(b² - 4ac\)

\[\begin{align}a &= 1,\;b= - 40,\;c = 400\\b^2 - 4ac& = {{( - 40)}^2} - 4(1)(400)\\& = 1600 - 1600\\& = 0\end{align}\]

Therefore, it is possible to design a rectangular park with the given condition:

\[\begin{align}x& = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\\

&= \frac{{ - b \pm 0}}{{2a}}\\& = \frac{{ - ( - 40)}}{{2(1)}}\\&= \frac{{40}}{2}\\&= 20\end{align}\]

\(b = 20,\, l = 40-b = 20\)

\(\therefore\;\)Yes, possible to design a rectangular park with side \( = 20\,\rm{m.}\)