Calculate the adjacent sides of a rectangle with a given perimeter as 100 cm and enclosing the maximum area.

$x = 25\text{ cm}, y = 25\text{ cm}$

The correct answer is Option (2) → $x = 25\text{ cm}, y = 25\text{ cm}$ ##

Let $x$ and $y$ be the adjacent sides of the rectangle,

$∴2x + 2y = 100 \Rightarrow x + y = 50 \dots(i)$

Let $A$ be the area of rectangle.

$∴A = xy$

Using (i), we get $A = x(50 - x)$

$\Rightarrow A = 50x - x^2$

$∴\frac{dA}{dx} = 50 - 2x$

For maximum area $\frac{dA}{dx} = 0 \Rightarrow 50 - 2x = 0$

$\Rightarrow x = 25$

When $x = 25, y = 50 - 25 = 25$

Hence, adjacent sides are $x = 25 \text{ cm}$ and $y = 25 \text{ cm}$.