You are designing a rectangular poster to contain 50 in^2 of printing with a 4-in. margin at the top and bottom and a 2-in margin at each side. What overall dimensions will minimize the amount of paper used?

Markus Petty 2022-07-20 Answered
You are designing a rectangular poster to contain 50 in^2 of printing with a 4-in. margin at the top and bottom and a 2-in margin at each side. What overall dimensions will minimize the amount of paper used?
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Answers (1)

umgangistbf
Answered 2022-07-21 Author has 12 answers
Let the paper size be x inches in length and y inches in width.
the length of the printed space would be x-8 inches and width would be y-4 inches. Print area would thus be (x-8)(y-4)= 50.
From this y = 4 + 50 x - 8 = 4 x + 18 x - 8 .
Also from the same equation on simplifying, it is xy-8y-4x+32=50.
Since the area of the paper of size x inches by y inches is xy, let it be denoted as A. Thus
A-8y-4x=18 Or A= 8y+4x+18
A = 32 x + 144 x - 8 + 4 x + 18 . For minimum paper size d A d x must be =0, hence,
d A d x = 32 ( x - 8 ) - ( 32 x + 144 ) ( x - 8 ) 2 + 4 = 0
- 400 ( x - 8 ) 2 + 4 = 0
( x - 8 ) 2 = 100
x-8=10
x=18, hence y= 4 + 50 x - 8 =9
Dimension of minimum paper size would be 18 inches by 9 inches.
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