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

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## 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.

Video Solution
Ex 4.4 | Question 5

## 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 $$= 2(l + b) = 80 \ldots \ldots (1)$$

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

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 $$= 20\,\rm{m.}$$

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