A book designer has decided that the pages of a book should...

A book designer has decided that the pages of a book should have 1-in. margins at the top and bottom and $\frac{1}{2}$ -in. margins on the sides. She further stipulated that each page should have an area of 50 in. $^{2}$ (see the accompanying figure). Determine the page dimensions that will result in the maximum printed area on the page.

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Key Concepts

Geometric Modeling and Margins

This concept relates to understanding how an initial geometric shape, such as a page with given overall dimensions, is partitioned into subregions (printed area and margins). It involves subtracting the dimensions taken by the margins from the total page dimensions to derive the size of the printed area, thereby linking geometry with optimization.

Derivative-Based Optimization

The use of calculus in optimization entails taking the derivative of the objective function with respect to the chosen variable, setting it equal to zero, and solving for the variable. This provides critical points that are examined to determine the maximum or minimum value of the function.

Optimization under Constraints

This concept involves formulating an objective function (in this case, the printed area) that needs to be maximized or minimized while satisfying a given constraint (the total page area). It often requires expressing one variable in terms of others using the constraint and then performing optimization using calculus techniques.

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