What should the dimensions of the poster be so that the least amount of poster is used?

Question

A rectangular poster is to contain 338 square inches of print. The margins at the top and bottom of the poster are to be 2 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the poster be so that the least amount of poster is used?

Answer 1

Length is L. Width is W.

LW = 338

W = 338/L

Amount Used = Total Area = A = (L + 4)(338/L + 2) - those are the length and width with the margins added.

FOIL: A = 338 + 2L + 1342/L + 8 = 2L + 1342/L + 346 = 2L +1342L^-1 + 346

A' = 2 -1342L^(-2) = 0

2L^2 -1342 = 0

L^2 = 1342/2 = 671

L = sqrt(671) = about +/- 25.9037

-inf to -25.9037: use -100: Derivative is positive. -25.9037 isn't a minimum

-25.9037 to 25.9037: Use 0. Derivative is negative

25.9037 to infinity: use 100: Derivative is positive.

Down before 25.9037, up after 25.9037. 25.9037 is a minimum.

W = 338/sqrt(671) = about 13.0483