Question

The adjacent sides of a rectangle with given perimeter as 100 cm and enclosing maximum area are

• A. 10 cm and 40 cm
• B. 20 cm and 30 cm
• C. 25 cm and 25 cm
• D. 15 cm and 35 cm

Answer 1

Answer: (C)

25 cm and 25 cm

Answer 2

$2x + 2y = 100$ ⇒ $x + y = 50$ .....(i)

Let area of rectangle is A, $\therefore$ $A = xy$ ⇒ $y = \frac{A}{x}$

From (i), $x + \frac{A}{x} = 50$ ⇒ $A = 50x - x^{2}$ ⇒ $\frac{dA}{dx} = 50 - 2x$

for maximum area $\frac{dA}{dx} = 0$

$\therefore$ $50 - 2x = 0$ ⇒ $x = 25$ and $y = 25$

$\therefore$ adjacent sides are 25 cm and 25 cm.