Johnny is designing a rectangular poster to contain 48inches squared of printing with a 4-in margin at the top and bottom and a 3-in margin at each side. What overall dimensions will minimize the amount of paper used?
Let the dimensions of the poster be x by y inches.
"xy=48\\to y=\\frac{48}{x}."
"A=(x+6*2)(y+8*2)=xy+16x+12y+192=16x+\\frac{576}{x}+240."
"\\frac{dA}{dx}=0\\to16-\\frac{576}{x^2}=0\\to x^2=36."
"x=\\sqrt{36}=6."
"y=\\frac{48}{6}=8."
"x+6*2=6+12=18."
"y+8*2=8+16=24."
The overall dimensions of the paper: 18 by 24 inches.
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