To find the dimensions of the poster with the smallest area, we can set up an equation and optimize it. Let's assume the length of the poster is L and the width of the poster is W.
From the given information, we know that the top and bottom margins are both 8 cm. So the effective length of the printed material will be L-2*8 = L-16.
Similarly, the side margins are each 9 cm. So the effective width of the printed material will be W-2*9 = W-18.
The area of the printed material is fixed at 382 square centimeters. This can be given as:
(L-16) * (W-18) = 382
Now, we need to minimize the area of the poster, which is given by A = L * W.
We can rewrite the area expression in terms of L, W:
A = (L-16) * (W-18) + 2 * 8 * (W-18) + 2 * 9 * (L-16)
Simplifying the equation:
A = L * W - 16W - 18L + 288 + 16W - 288 + 18L - 288
A = L * W - 16W - 18L + 16W - 18L
A = L * W - 34L - 34W
Now, we need to find the values of L and W that minimize the area A. To do this, we can take partial derivatives of A with respect to L and W, and set them equal to zero:
∂A/∂L = W - 34 = 0 (partial derivative with respect to L)
∂A/∂W = L - 34 = 0 (partial derivative with respect to W)
Solving these equations, we get L = W = 34.
Therefore, the dimensions of the poster with the smallest area are 34 cm for both the length and width.