Question

A rod $108$ metres long is bent to form a rectangle. Find its dimensions, if its area is maximum.

Answer 1

Length of the rod $L=108m$

Let side of the rectangle be ${}^{\prime }{x}^{\prime }$ and ${}^{\prime }{y}^{\prime }$.

Perimeter

$=2x+2y=108$

$x+y=54$

$y=54-x$

Area $=xy$

$A=x(54-x)$

$A=54x-x^2$ $...........\left(1\right)$

On differentiating w.r.t $x$, we get

$\frac{dA}{dx}=54-2x$

For maximum, $\frac{dA}{dx}=0$

$54-2x=0$

$x=27$

On differentiating w.r.t $x$, we get

$\frac{d^{2}A}{d{x}^2}=0-2=-2$

$\frac{d^{2}A}{d{x}^2}=-2<0$

So, Area is maximum.

Now, Dimension $x=27m$

$y=54-27=27m$

Hence, this is the answer.