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A sheet of paper for a poster is 1.5m2 in area . The margins at the top and the bottom are to be 6cm wide at the sides , 4cm wide . Whatr should be the dimensions of the sheet to maximize the printed area ?

Please help soon.

Thank you.



Let the length of the sheet = x cmLet the breadth of the sheet = y cmNow, area of the sheet = 1.5 m2 = 15000 cm2⇒xy = 15000⇒y = 15000x    ......1Now, length of the printed portion = x - 12 cmbreadth of the printed portion = y - 8 cmNow, area of printed portion =  x - 12y - 8⇒A = xy - 8x - 12y + 96⇒A = 15000 - 8x - 12 × 15000x + 96⇒A = 15096 - 8x - 180000x⇒dAdx = -8 + 180000x2For maxima or minima,dAdx = 0⇒-8 + 180000x2 = 0⇒8x2 = 180000⇒x2 = 22500⇒x = 150Now,d2Adx2 = -360000x3d2Adx2x=150 = -360000150×150×150 = -875<0So, printed area is maximum at x = 150Now, from 1, y = 15000150 = 100So, length of the sheet = 150 cmbreadth of the sheet = 100 cm

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