Final answer:
The optimization problem seeks to find the values of width (W) and height (H) of a page that maximize the print area, given that the total area is 96 square inches and margins are specified. Calculus can be used to solve for the dimensions by expressing the height in terms of the width and taking the derivative of the print area to find the maximum value.
Step-by-step explanation:
To find the dimensions of a page that maximize the print area with given margins, we first establish the constraints based on the information provided. The print area is equal to the area of the page minus the area taken up by the margins. Given that we have 1 inch margins on either side and 1 1/2 inches at the top and bottom, we can express the dimensions of the print area as width (W) and height (H), which are related to the dimensions of the page.
Therefore, the print area A can be expressed as:
A = (W - 2 * 1 inch) * (H - 2 * 1.5 inches)
Since the total area of the page is 96 square inches, we have:
W * H = 96 square inches
To maximize the print area, we need to find the values of W and H that satisfy both equations. We can use the second equation to express H in terms of W:
H = 96 / W
Plugging this back into the print area equation, we can express A in terms of W:
A = (W - 2) * ((96 / W) - 3)
To find the maximum value of A, we can take the derivative with respect to W, set it to zero, and solve for W. This optimization problem uses the principles of calculus to determine the most efficient dimensions.