Question

A rectangular page is to contain 24square inches of print. The margins at the top and bottom of the page are to be 1.5 inches each and the margins on the left and right are to be 1 inch each. What should the dimensions of the rectangular page be so that the least amount of paper is used?

Answer 1

Ai = Li*Wi = 24in²
A = L*W
A = (Li+2*1.5)(Wi+2*1)
A = (Li+3)(Wi+2)
A = (Li+3)((Ai/Li)+2)
A = 2Li+3Ai/Li+(Ai+6)

Find minimum length by setting derivative =0:
dA = 2-3Ai/(Li)² = 0
Li = √(3Ai/2) = 6in
L = Li+3 = 9in
Wi = Ai/Li = 4in
W = Wi+2 = 6in

So, 6x9 is smallest page size to hold print inside 4x6 print area with margins.