Mathematics
Is it possible to design a rectangular park of perimeter 80 m and area 400 m2 ? If so, find its length and breadth.
Quadratic Equations
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Answer
Let length of rectangular park be l and breadth be b meters.
Given,
Perimeter = 80 meters.
⇒ 2(l + b) = 80
⇒ l + b = 40
⇒ b = 40 - l ………..(1)
Given,
⇒ Area = 400 m2
⇒ lb = 400
⇒ l(40 - l) = 400
⇒ 40l - l2 = 400
⇒ l2 - 40l + 400 = 0
Comparing l2 - 40l + 400 = 0 with ax2 + bx + c = 0, we get :
a = 1, b = -40 and c = 400.
Substituting value in b2 - 4ac, we get :
⇒ (-40)2 - 4 × 1 × 400
⇒ 1600 - 1600
⇒ 0.
Since, b2 - 4ac = 0.
∴ Roots are real and equal.
Solving, l2 - 40l + 400 = 0, we get :
⇒ l2 - 20l - 20l + 400 = 0
⇒ l(l - 20) - 20(l - 20) = 0
⇒ (l - 20)(l - 20) = 0
⇒ l - 20 = 0
⇒ l = 20 meters.
Breadth (b) = 40 - l = 40 - 20 = 20 meters.
Hence, condition is possible and length = 20 meters and breadth = 20 meters.
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