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 -

Answer 1

Given:

Sides of a rectangle = a and b units

To find:

Relation between a and b for which area of the rectangle is maximum.

Solution:

Since, perimeter (P) of a rectangle is,

P = 2(a + b)

a = $\frac{P}{2}-b$

Area (A) of a rectangle = Length × Width

A = a × b

A = $(\frac{P}{2}-b)b$

A = $\frac{Pb}{2}-b^{2}$

Since, area (A) of the given rectangle is maximum,

$\frac{dA}{db}=0$

$\frac{dA}{db}=\frac{P}{2}-2b=0$

b = $\frac{P}{4}$

Since a = $\frac{P}{2}-b$

a = $\frac{P}{2}-\frac{P}{4}$

a = $\frac{P}{4}$

Therefore, a = b = $\frac{P}{4}$

And the relation between a and b is,

a = b