Some years from now you are working for a book publisher. Your boss asks you to give him a formula that will tell him the length and width of a book page that contains A square inches of printed text, a left margin of L inches, a right margin of R inches, a top margin of T inches, and a bottom margin of B inches, and that otherwise has an area as small as possible. After dusting off your Calculus notes you tell him that the length of that page equals _________ inches, and the width equals _________ inches. Of course, both of your answers are in terms A, L, R, T, and B

Answer 1

If the printed area has dimensions x and y, then xy=A, so y=A/x

The actual page has dimensions
x+L+R and y+T+B

To minimize the page area, then you want to minimize

f(x,y) = (x+L+R)(y+T+B)
f(x) = (x+L+R)(A/x + T+B)
= (T+B)x + A + A(L+R)/x + (L+R)(T+B)

df/dx = (T+B) - A(L+R)/x^2
df/dx = 0 when x^2 = A(L+R)/(T+B)

. . .