The dimensions of the poster with the smallest area are 20 cm by 42 cm.
Let x and y be the width and height of the printed material on the poster, respectively.
Then the overall width of the poster is x + 12 and the overall height of the poster is y + 18.
We are given that the area of the printed material on the poster is fixed at 864 cm², so we have the equation xy = 864.
We want to find the dimensions of the poster with the smallest area.
This means we want to minimize the perimeter of the poster, which is given by P = 2(x + 12) + 2(y + 18) = 2x + 2y + 60.
Substituting the equation xy = 864 into the expression for P, we get P = 2x + 2y + 60 = 2y + 2(864/y) + 60.
This is a function of one variable (y), so we can minimize it using calculus.
Taking the derivative of P with respect to y, we get P'(y) = 4 - 1728/y². Setting P'(y) = 0, we get y² = 1728.
Solving for y, we find y = 42.
Substituting this back into xy = 864, we find x = 20.
Therefore, the dimensions of the poster with the smallest area are 20 cm by 42 cm.