Answer to Question #129848 in Calculus for Hetisani Sewela

Question #129848

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?


1
Expert's answer
2020-08-18T16:48:07-0400

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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