Let l represent the length of the printed rectangular region of the page.
Given that the area of the printed rectangular region is 36, then
width of the printed portion or region = 36/l
The margin left on both sides is 1.5 inches. Thus,
length of page = l + 1.5(2) = l + 3
width of page = 36/l + 1.5(2) = 36/l + 3
Area = length x width
Area = (l + 3)(36/l + 3)
Area = 36 + 3l + 108/l + 9
Area = 36 + 9 + 3l + 108/l
A(l) = 45 + 3(l + 36/l)
We would minimise A(l)
For Amin, A'(l) = 0
3(1 - 36/l^2) = 0
3 = 0 or 1 - 36/l^2 = 0
1 = 36/l^2
l^2 = 36
l = ±√36
l = ±6
Also,
A''(l) > 0
A''(l) = 3(0 - 36(-2)l^-3) = 72/l^3
Substituting l = 6,
72/6^3 > 0
Thus,
l = 6 gives Amin
The dimensions would be
length = l = 6 + 3 = 9
width = 36/6 + 3 = 6 + 3 = 9
Length = 9 inches
width = 9 inches
