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 of given rectangle, $A=a\times b$ Convert area in terms of 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, $\frac{dA}{da}=0$ & $\frac{d^{2}A}{d{a}^2}<0$ $\frac{dA}{da}=\frac{p}{2}-2a=0$ $a=\frac{p}{4}\Rightarrow b=\frac{p}{4}\Rightarrow a=b$