Question

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

• A. $a+b=0$
• B. $a=b$
• C. $a.b=1$
• D. $a=4b$

Answer 1

The correct option is B $a=b$

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

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

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