Q.18. A sheet of paper is to contain 18  c m 2 of printed matter. The margins at the top and bottom are 2 cm each, and at the sides 1 cm each. Find the dimensions of the sheet which require the least amount of paper.

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Le the length of the sheet= x cmBreadth of the sheet= y cmNow, area of sheet=18 cm2xy=18y=18x  ...1Now, length of printed portion=x-4 cmBreadth of printed portion=y-2 cmNow area of printed portion=x-4y-2A=xy-2x-4y+8A=18-2x-418x+8A=18-2x-72x+8dAdx=-2+72x2For maxima and minimadAdx=0-2+72x2=0-2x2+72=0x2=36x=6Now,d2Adx2=-144x3d2Adx2x=6=-14463=-23<0So, printed area is maximum at x=6Now, from (1), y=186=3So, length of the sheet=6cmbreadth of the sheet=3cm

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