Q5:

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

Given:

Perimeter of the rectangular park ($P$) = $80$ m.

Area of the rectangular park ($A$) = $400$ m$^2$.

To Find:

Determine if such a park can be designed, and if so, find its length ($l$) and breadth ($b$).

Step 1: Formulating the Equations

Let the length of the rectangular park be $l$ meters and the breadth be $b$ meters.

The formula for the perimeter of a rectangle is $P = 2(l + b)$.

Given $P = 80$, we have:

$2(l + b) = 80$

$l + b = 40$

$b = 40 - l$ --- (Equation 1)

The formula for the area of a rectangle is $A = l \times b$.

Given $A = 400$, we have:

$l \times b = 400$ --- (Equation 2)

Step 2: Substituting Equation 1 into Equation 2

Substitute $b = 40 - l$ into the area equation:

$l(40 - l) = 400$

$40l - l^2 = 400$

Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:

$l^2 - 40l + 400 = 0$

Step 3: Checking the Discriminant

To determine if the roots are real, we calculate the discriminant ($D = b^2 - 4ac$).

Here, $a = 1$, $b = -40$, and $c = 400$.

$D = (-40)^2 - 4(1)(400)$

$D = 1600 - 1600$

$D = 0$

[Since $D = 0$, the quadratic equation has two equal real roots, meaning it is possible to design such a park.]

Step 4: Solving for $l$

Using the quadratic formula $l = \frac{-b \pm \sqrt{D}}{2a}$:

$l = \frac{-(-40) \pm \sqrt{0}}{2(1)}$

$l = \frac{40}{2}$

$l = 20$ m

Step 5: Finding the breadth $b$

Substitute $l = 20$ into Equation 1:

$b = 40 - 20$

$b = 20$ m

Conclusion:

Since the length and breadth are equal, the rectangular park is a square with side length $20$ m.

Final Answer: Yes, it is possible to design the park. The length is 20 m and the breadth is 20 m.