Answer to Question #295632 in Calculus for abhii

Question

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 2.5 cm

wide margin along the top and bottom and 2.0 cm wide margin along the sides. What

are the most economical dimensions of each printed page?

Answer 1

Let $x+4=$ the width dimension of total page in cm, let $y+5=$ the length dimension of total page in cm.

Given $xy=180$ cm².

The total area $A$ of the page will be $A=(x+4)(y+5).$

Substitute $y=\dfrac{180}{x}$

$$A=A(x)=(x+4)(\dfrac{180}{x}+5) ,x>0$$

$$A(x)=200+5x+\dfrac{720}{x}$$

Differentiate with respect to $x$

$$A'(x)=5-\dfrac{720}{x^2}$$

Find the critical number(s)

$$A'(x)=0=>5-\dfrac{720}{x^2}=0$$

$$x^2=144$$

$$x_1=-12, x_2=12$$

We consider $x>0$

If $0<x<12, A'(x)<0, A(x)$ decreases.

If $x>12, A'(x)>0, A(x)$ increases.

The function $A(x)$ has a local minimum at $x=12.$

Since the function $A(x)$ has the only extremum for $x>0,$ then the function $A(x)$ has the absolute minimum for $x>0$ at $x=12.$

Then $y=\dfrac{180}{12}=15$

 The most economical dimensions of each printed page are

$$wigth\times length=16cm\times 20cm$$