Question

Is it possible to design a rectangular park of perimeter $80m$ and area $400m^2$? If so find its length and breadth.

Answer 1

Step $1$: Make a mathematical equation using giving conditions :

Let the length and breadth of the park is $l$ and $b$.

Given: Perimeter of rectangular park is $80m$.

$$\begin{gathered} \Rightarrow 2\left(l+b\right)=80 \\ \Rightarrow l+b=40 \\ \Rightarrow b=40-l \end{gathered}$$

We know that, Area $\left(A\right)$ of the rectangle is given by :

$$\begin{gathered} \Rightarrow A=l\times b \\ \Rightarrow A=l(40-l) \\ \Rightarrow 400=40l-l^2 \\ \Rightarrow l^2-40l+400=0 \end{gathered}$$`

Step $2$: Compute the discriminant of the quadratic equation.

Comparing the quadratic equation with $a{x}^2+bx+c=0$,

$a=1,b=-40,c=400$

We know that, Discriminant$\left(D\right)$$=b^2-4ac$

$\Rightarrow$ $D={(-40)}^2-4\left(1\right)\left(400\right)$

$\Rightarrow$ $D=1600-1600$

$\Rightarrow$ $D=0$

Hence, this equation has equal roots.

Step $3$: Compute the roots of the equation.

For equal roots, roots are given by,

Roots of the equation$=-\frac{b}{2a}$

$\Rightarrow$Roots of the equation$=-\frac{(-40)}{2}$

$\Rightarrow$Roots of the equation$=20$

Hence, the value of $l$ is $20m.$

Step $4$: Compute the length and breadth of the park.

Length $\left(l\right)=20m$

Breadth $\left(b\right)=40-20$

Breadth $\left(b\right)=20m$

Hence, the length of the park is $20m$ and the breadth of the park is $20m$.