A rectangular page is to contain 180 square inches of print. The top and bottom margins are each 0.8 inch wide, and the margins on each side are 1.25 inch wide. what should the dimensions be if the least amount of material is to be used?
The rectangular page is to contain 180 square inches of area that can be printed on, the top and bottom margins are 0.8 inch wide and the side margins are 1.25 inch wide.
Let the length of the side of the area to be printed on be L, the length of the top (and bottom) is 180/L. The area of the page is A = (180/L + 2.5)*(L + 1.6) = 180 + 2.5L + 288/L + 4
To minimize the area solve `(dA)/(dL)` = 0 for L
=> `2.5 - 288/L^2 = 0`
=> `288/L^2 = 2.5`
=> `L^2 = 288/2.5`
=> `L = 24/sqrt 5 ~~ 10.73 ` inch
The length of the side should be `(15*sqrt 5)/2 ~~ 16.77 ` inch
The dimensions of the page should be approximately 16.77 inch x 10.73 inch.