The production manager of a company plans to include 180 sq. cm of actual printed matter in each page of a book under production. Each page should have a 2.5 cm wide margin along the top and bottom and 2 cm wide margin along the sides. What are the most economical dimensions of each printed page?

16 cm × 20 cm

The correct answer is Option (3) → 16 cm × 20 cm

Let $x\, cm (x > 0)$ be the one dimension of printed matter, then the other dimension of printed matter is $\frac{180}{x}$ cm because the area of the actual printed matter is given to be 180 sq. cm.

Let A $cm^2$ be the area of the page, then

$A = (x+4)(\frac{180}{x} + 5) = 180+5x+\frac{720}{x}+20$

Differentiating w.r.t. x, we get

$\frac{dA}{dx}=0+5-\frac{720}{x^2}+0$ and $\frac{d^2A}{dx^2}=\frac{1440}{x^3}$

Now, $\frac{dA}{dx}=0⇒5=\frac{720}{x^2}⇒x^2=144⇒x=12$

Also, $(\frac{d^2A}{dx^2})_{x=12}=\frac{1440}{1728}>0$ ⇒ A is minimum when $x = 12$.

Hence, the most economical dimensions of the each printed page is $x+4=12+4=16\,cm$ and $\frac{180}{x}+5=\frac{180}{12}+5=20\,cm$.