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Q3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800\(m^2\). If so, find its length and breadth.
Answer :

Ans.Let breadth of rectangular mango grove = x metres
Let length of rectangular mango grove = 2x metres
Area of rectangle = length × breadth = \(x × 2x = 2x^2 m^2\)
According to given condition:
\(2x^2 = 800\)
\(2x^2 - 800 = 0 => x^2 - 400 = 0\)
Comparing equation \( x^2 - 400 = 0\) with general form of quadratic equation \(ax^2 + bx + c = 0\) ,
we get a = 1, b = 0 and c = -400
Discriminant = \( b^2 -4ac = (0)^2 - 4 (1) (-400) = 1600\)
Discriminant is greater than 0 means that equation has two distinct real roots.
Therefore, it is possible to design a rectangular grove.
Applying quadratic formula,\(x = {{-b ± \sqrt{b^2 - 4ac}} \over {2a}}\) to solve equation,
\(x = {{0 ± \sqrt{1600}} \over {2(1)}}\) = ± {{40} \over {2} } = ± 20\)
x = 20, -20
We discard negative value of x because breadth of rectangle cannot be in negative.
Therefore, x = breadth of rectangle = 20 metres
Length of rectangle = 2x = 40 metres