Final answer:
To find the dimensions of the poster with the smallest area, we can consider the printed material as a rectangle and set up an equation using the given information. One approach is to minimize the product of the dimensions using either graphing or calculus. Alternatively, we can simplify the equation to find the dimensions directly.
Step-by-step explanation:
To find the dimensions of the poster with the smallest area, we can consider the printed material as a rectangle with length and width.
Let the length of the printed material be x and the width be y.
Since the top and bottom margins are each 12 cm and the side margins are each 8 cm, the length of the poster would be x + 2(12) = x + 24 cm and the width would be y + 2(8) = y + 16 cm.
Given that the area of the printed material is fixed at 1536 cm², we have the equation:
(x + 24)(y + 16) = 1536
To find the dimensions with the smallest area, we need to minimize the product (x + 24)(y + 16).
One approach is to find the value of x that minimizes the product for different values of y, and then find the value of y that minimizes the product for the values of x obtained.
This can be done by graphing the product as a function of x and y, or by using calculus to find the critical points of the function.
Alternatively, we can simplify the equation (x + 24)(y + 16) = 1536 to find the dimensions of the poster with the smallest area. After that, we can substitute these values back into the equation to verify that they satisfy the equation.