A billboard designer has decided that a sign advertising the movie Fight Club II - Rage of the Mathematicians should have 1-ft. margins at the top and bottom, and 2-ft. margins on the left and right sides. Furthermore, the billboard should have a total area of 200 ft2 , including the margins. Find the dimensions that would maximize printed area.

Answer:Length = 20 ftHeight = 10 ftStep-by-step explanation:Let 'X' be the total width of the billboard and 'Y' the total height of the billboard. The total area and printed area (excluding margins) are, respectively:[tex]200 = x*y\\A_p = (x-4)*(y-2)[/tex]Replacing the total area equation into the printed area equation, gives as an expression for the printed area as a function of 'X':[tex]y=\frac{200}{x} \\A_p = (x-4)*(\frac{200}{x} -2)\\A_p=208 -2x -\frac{800}{x}[/tex]Finding the point at which the derivate for this expression is zero gives us the value of 'x' that maximizes the printed area:[tex]\frac{dA_p(x)}{dx} =\frac{d(208 -2x -\frac{800}{x})}{dx}=0\\0=-2 +\frac{800}{x^2} \\x=\sqrt{400}\\x=20\ ft[/tex]If x = 20 ft, then y=200/20. Y= 10 ft.The dimensions that maximize the printed area are:Length = 20 ftHeight = 10 ft