) the top and bottom margins of a poster are 4 cm and the side margins are each 5 cm. if the area of printed material on the poster is fixed at 390 square centimeters, find the dimensions of the poster with the smallest area.
If x represents the width of the poster (including borders), the area of the finished poster can be written as .. a = x*(390/(x -10) +8) .. = 8x +390 +3900/(x -10)
Then the derivative with respect to x is .. da/dx = 8 -3900/(x -10)^2 This is zero at the minimum area, where .. x = √(3900/8) +10 ≈ 32.08 . . . . cm The height is then .. 390/(x -10) +8 = 8 +2√78 ≈ 25.66 . . . . cm
The poster with the smallest area is 32.08 cm wide by 25.66 cm tall.
_____ In these "border" problems, the smallest area will have the same overall dimension ratio that the borders have. Here, the poster is 10/8 = 1.25 times as wide as it is high.
