Question

A printed page is to have a total area of $80$ sq. cm with a margin of $1$ cm at the top and on each side and a margin of $1.5$ cm at the bottom. What should be the dimensions of the page, so that the printed area will be maximum?

Answer 1

Let $x$ cm and $y$ cm be the dimensions of the printed page.
$\Rightarrow xy=80$ ......... (i)
Let $A$ ($x$ sq. cm) be the printed area then,
$A=(x-2)(y-5/2)$
$A=xy-\frac{5x}{2}-2y+5$
$A=80-\frac{5x}{2}-\frac{2\times 80}{x}+5$ [using (i)]
$A=85-\frac{5x}{2}-\frac{160}{x}$

Differentiating w.r.t. $x$, we get
$\frac{dA}{dx}=\frac{-5}{2}+\frac{160}{x^2}$

Differentiating once again w.r.t. $x$

$\frac{d^{2}A}{d{x}^2}=\frac{-320}{x^3}$

For Maximum and Minimum area, $\frac{dA}{dx}=0$

Therefore, $\frac{-5}{2}+\frac{160}{x^2}=0$
$\Rightarrow \frac{160}{x^2}=\frac{5}{2}$
$\Rightarrow x^2=64$
$\Rightarrow x=8$

Also, ${\left(\frac{d^{2}A}{d{x}^2}\right)}_{x=8}=-\frac{320}{512}<0$

$\Rightarrow$ $A$ is maximum at $x=8$

From (i), $xy=80$

$\Rightarrow 8\times y=80$
$\Rightarrow y=10$

Therefore, the dimensions of the printed page are $10$ cm and $8$ cm.