A poster is to be printed on a rectangular sheet of paper of area 18 ~m ^2 . The margins at the top and bottom of 75 ~cm each and at the sides 50 ~cm each are to be left. Then the dimensions i.e. height and breadth of the sheet, so that the space available for printing is maximum, are respectively.

Step-by-step solution

Let height and breadth of the sheet be ' y ' m and ' x ' m respectively. aligned & x y=180000 ~cm ^2 & y= 180000 x aligned The area available for printing is aligned A & =(y-150)(x-100) & = ( 180000 x -150 )(x-100) & =180000- 18000000 x -150 x-15000 & =165000-150 x- 18000000 x dA d x & =0-150+ 18000000 x^2 dA d x & =0 x^2= 18000000 150 =120000 x=200 3 ~cm & y= 180000 200 3 =300 3 ~cm aligned Now, d ^2 ~A ~d x^2 = -36000000 x^3 At x=200 3 ~cm , d ^2 ~A ~d x^2 < 0 Area is maximum at x=200 3 ~cm and aligned y & =300 3