Let's denote the width of the printed material on the poster as "x" cm and the height as "y" cm.
According to the given information, the top and bottom margins are each 6 cm, and the side margins are each 8 cm. This means that the actual width of the entire poster, including the margins, is "x + 2(8)" cm, and the actual height, including the margins, is "y + 2(6)" cm.
Given that the area of the printed material on the poster is fixed at 380 square centimeters, we can set up the following equation:
Actual Area of Poster = Area of Printed Material on Poster
(x + 2(8))(y + 2(6)) = 380
(x + 16)(y + 12) = 380
To find the dimensions of the poster with the smallest area, we need to minimize the product (x + 16)(y + 12).
Since the given area of the printed material on the poster is fixed at 380 square centimeters, the actual area of the entire poster, including the margins, will be minimized when (x + 16)(y + 12) is minimized.
To minimize the product (x + 16)(y + 12), we need to minimize both x + 16 and y + 12, as they are both positive quantities.
Since x and y represent the width and height of the printed material on the poster, respectively, the smallest possible values for x + 16 and y + 12 would be 0, which means x = -16 and y = -12. However, since width and height cannot be negative, we need to find the next best option.
The smallest possible values for x + 16 and y + 12 that are greater than or equal to 0 would be when x = 0 and y = 0. This means that the width of the printed material on the poster should be 0 cm and the height should be 0 cm, which would make the dimensions of the poster with the smallest area:
Width = 0 cm
Height = 0 cm
However, please note that this would mean there is no printed material on the poster, as the width and height are both 0. If you want to have a non-zero width and height for the printed material on the poster, you would need to adjust the given area of the printed material on the poster accordingly.