A graphic designer is asked to create a movie poster with a 65 square-inch photo surrounded by a 3 inch border at the top and bottom and a 2 inch border on each side. What overall dimensions for the poster should the designer choose to use the least amount of paper?
if the printed area is x inches wide and y inches high then we have the paper area is
a = (y+6)(x+4)
since xy=65, that gives us
a = (65/x + 6)(x+4) = 6x+89 + 260/x
a has a minimum when da/dx = 0, so we want
6 - 360/x^2 = 0
x = √(130/3)
y = √390/2
and the paper area is
a = (√390/2+6)(√(130/3)+4) = 89+4√390