A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increase by 68 cm\(^2\) if we add 2 cm to each side of the rectangle. Find the length of the shorter side.
Let the length of the longer side = \(x\) cm
∴ The length of the shorter side = (\(x\) - 6) cm
If we increase each side's length by 2 cm, it becomes
(\(x\) + 2) cm and (\(x\) - 4) cm respectively
Area of a rectangle = L x B
\(A_1 = x(x - 6) = x^2 - 6x\)
\(A_2 = (x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8\)
\(A_1 + 68 = A_2\) (Given)
⇒ \(x^2 - 6x + 68 = x^2 - 2x - 8\)
⇒ \(x^2 - x^2 - 6x + 2x\) = -8 - 68
⇒ -4\(x\) = -76
⇒ \(x\) = \(\frac{-76}{-4}\) = 19cm
∴ The length of the shorter side = \(x\) - 6 = 19 - 6 = 13 cm
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