What are the dimensions of the largest printed area?

Question

A printer needs to make a poster that will have a total area of 200 in² and will have 1 in margins on the sides, a 2 in margin on the top and a 1.5 in margin of the bottom. What dimensions of the poser will give the largest printed area

Answer 1

Hi Diego,

 

let x = length of the full poster

 

then 200/x = width of the full poster

 

so

 

length of the printed area is x - 3.5

 

and width of the printed area is (200/x)-2

 

so

 

Area of Printed space = (x-3.5)((200/x)-2)

 

take the derivative using the product rule

 

A' = (x-3.5)(-200/x²) + ((200/x)-2)

 

A' = 700 -2x²       (you can fill in the steps - I got lazy)

 

now set the derivative to o

 

0 = 700 -2x²

 

700 = 2x²

 

350 = x²

 

x = 18.70829  The original length

 

width = 10.69045

 

area of space available for printing is

 

15.20829  *  8.69045

 

= 132.1669 sq in

 

Hope this helps

 

 

 

 

Answer 2

Slightly Easier Calculation

Width * Height = area. The width has 2 added for the margins. The height has 3.5 added for the margins.

w is width of the full poster.

h is height of the full poster = 200/w

w of the printed area = w-2 (subtract the margins).

h of the printed area = 200/w - 3.5 (subtract the margins top and bottom)

Printed Area = A = (w-2)(200/w - 3.5) = 200 - 400/w - 3.5w + 7 = -3.5w -400/w + 207

A' = -3.5 -400/w^2 = 0

-3.5w^2 -400 = 0

w^2 = 400/3.5

w = sqrt (400/3.5) = 10.6904

Height = 200/sqrt (400/3.5) = 18.7083