Question

You are designing a rectangular poster to contain 256 in^2 of printing with a 3-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?

Answer 1

let the width of the paper be x in
let the height of the paper be y in

then the width of the printed area is x-4 and the height of the printed area is y-5

we know (x-4)(y-5) = 256
xy - 5x - 4y + 20 = 256
y(x-4) = 5x + 236

y = (5x+236)/(x-4)

Area of page
= xy
= x(5x+236)/(x-4)
= (5x^2 + 236x)/(x-4)

Now differentiate that using the quotient rule, set it equal to zero and solve for x
note that x>4 to make sense, sub your x value back into the above y = ... relation to get the dimensions.