You are designing a poster to contain 50 cm2 of printing with margins of 4 cm each at the top and bottom and 2 cm at each side. What overall dimensions will minimize the amount of paper used?
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+\\frac {50} {X-8}=\\frac{4X+18}{X-8}"
Also from the same equation on simplifying, it is "XY-8X+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=\\frac{32X+144}{X-8}+4X+18" . For minimum paper size "\\frac{dA}{dX}" must be "=0" , hence,
"\\frac{dA}{dX}=\\frac{32(X-8)-(32X+144)}{(X-8)^2}+4=0"
"\\frac{-400}{(X-8)^2}+4=0"
"(X-8)^2=100"
"X-8=10"
"X=18" , hence "Y=4+\\frac{50}{18-8}=9"
Dimension of minimum paper size would be 18 inches by 9 inches.
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