Question

If sides of a rectangle with given perimeter $p$ are $x$ & $y$, then find the relation between $x$ & $y$ for which area of the given rectangle is maximum.

• A. $x+y=0$
• B. $x=y$
• C. $x.y=1$
• D. $x=4y$

Answer 1

The correct option is B $x=y$

Perimeter p of rectangle having side length $x$ and $y$ is
$p=2(x+y)=$ constant
Area $A$ of given rectangle is
$A=x\times y$

Convert area in single variable $x$ using perimeter equation,
$A=x\times y=x\times (\frac{p}{2}-x)=x\frac{p}{2}-x^2$

For area to be maximum,
$\frac{dA}{dx}=0$ & $\frac{d^{2}A}{d{x}^2}<0$
$\frac{dA}{dx}=\frac{p}{2}-2x=0$
$x=\frac{p}{4}\Rightarrow y=\frac{p}{4}\Rightarrow x=y$