Final answer:
To minimize the poster's area with fixed printed material at 1536 cm², the printed area should be a square, leading to a calculation of the square root of 1536 for both dimensions, and adding the respective margins to obtain the total poster dimensions.
Step-by-step explanation:
To find the dimensions of the poster with the smallest area that fits the given area of printed material, we first denote the width and height of the printed area as x and y respectively (in centimeters). Given the area of the printed material is 1536 cm², we set up the equation x×y = 1536.
Considering the margins, the total width of the poster would be x + 2×8 (for both side margins), and the height would be y + 2×12 (for the top and bottom margins). To minimize the area of the poster, we use the method of Lagrange multipliers or optimize the area function A(x, y) = (x+16)(y+24) with the constraint x×y = 1536.
Through optimization, we find that the dimensions of the printed material that give the smallest poster area when accounting for the margins are when x and y are equal, signifying a square shape for the printed area. So, if the printed area is a square, x = y = √1536 cm. Then, the total dimensions of the poster are (width × height): (√1536+16) cm by (√1536+24) cm.