Is it possible to design a rectangular park with a perimeter of 80 and an area of 400 m^{2} ? If

Question 5 Is it possible to design a rectangular park with a perimeter of 80 and an area of 400 $ m^{2}$ ? If so, find its length and breadth.

Solution:

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

Perimeter of the rectangular park = $2 (l + b)$ = 80

So, $l + b$ = 40

Or, b = $40 – l$

Area of the rectangular park = $l×b$ =$ l(40 – l)$ = $40l – l^{2} $ = 400

$ l^{2} – 40l + 400 $= 0, which is a quadratic equation.

Comparing the equation with $ ax^{2} + bx + c $ = 0, we get

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

Since discriminant = $ b^{2} – 4ac$

=$ (-40)^{2} – 4 × 400$

= 1600 – 1600 = 0

Thus, $ b^{2} – 4ac$ = 0

Therefore, this equation has equal real roots. Hence, the situation is possible.

The root of the equation,

$l$ = $ \dfrac{–b}{2a}$

$l$ =$ \dfrac{ -(-40)}{2(1)}$ = $ \dfrac{40}{2}$ = 20

Therefore, the length of the rectangular park, $l $= 20 m

And the breadth of the park, b = 40 –$ l$ = 40 – 20 = 20 m.

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