Final answer:
To determine the dimensions of the poster for the maximum printing area, you need to create equations for the perimeter and the printable area, taking into account the clearances. The perimeter constraint gives you one equation, and you can use this to find the relationship between the length and width, then substitute into the area expression to optimize it.
Step-by-step explanation:
The overall dimensions of the poster to maximize the printing area, given a maximum perimeter of 42 inches and a required clearing from the edges, can be determined using algebra. Let's denote the length of the poster as L inches and the width as W inches. Because of the clearings, the printable area length will be L - 3 inches (subtracting 1.5 inches from each side) and the printable area width will be W - 2 inches (subtracting 1 inch from top and bottom).
The perimeter of the poster can be represented as 2L + 2W = 42. To maximize the printable area, we want to maximize the quantity (L - 3)(W - 2), which represents the area. After finding W from the perimeter equation (W = 21 - L), we can substitute it into the area equation and maximize the quadratic function. Using the vertex form of a quadratic, we can determine the dimensions that give the maximum area. Typically, a quadratic function will be maximized when L and W are equal, assuming the shape is a rectangle, leading to a square as the most area-efficient shape. However, due to the edge clearances not being equal, the optimized dimensions may be slightly different, and you can use calculus or other optimization techniques to find the exact maximum dimensions.