Each page of a book will contain 30 in squared of print, and each page must have 2 inch margins at top and bottom and 1 inch margin at each side. What is the minimum possible area of such a page.

Question

Each page of a book will contain 30 in squared of print, and each page must have 2 inch margins at top and bottom and 1 inch margin at each side.  What is the minimum possible area of such a page.

Steve · Answer

if the page width = w, the height = 30/w
Allowing for two margins on each dimension,

a(w) = (width + 2*2)(30/w + 2*1)
= (w+4)(30/w+2)
= 30 + 2w + 120/w + 8
= 2w + 120/w + 38

area is min where da/dw = 0

da/dw = 2 - 120/w^2
2w^2 = 120
w^2 = 60
w = 7.75

a(7.75) = 68.98 -- call it 69 in²