Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 \(50 \mathrm{in} .{ }^{2}\) (see the accompanying figure). Determine the page dimensions that will result in the maximum printed area on the page.

Step-by-Step Solution

Answer

The dimensions of the printed area that will result in the maximum printed area are width x=25 inches and length y=1 inch.

Step 1: Define variables

Let x be the width of the printed area and y be the length of the printed area (both in inches).

Step 2: Write the equation for the page area

Taking the margins into account, the total width of the page is (x + 1) inches and the total length of the page is (y + 1) inches. The area of the page is given by: \(A = (x + 1)(y + 1)\) Since the area of each page is 50 square inches, the equation becomes: \(A = (x + 1)(y + 1) = 50\)

Step 3: Solve for y

Solve for y in terms of x: \(y+1 = \frac{50}{x+1}\) \(y = \frac{50}{x+1} - 1\)

Step 4: Find the printed area function

The printed area of the page P is the product of the width and length of the printed area: \(P(x) = xy\) Replacing y with the expression obtained in Step 3: \(P(x) = x\left(\frac{50}{x+1} - 1\right)\)

Step 5: Find the critical points

To find the critical points, we must find the derivative of the printed area function P(x) and set it equal to zero: \(P'(x) = \frac{d}{dx}(x(\frac{50}{x+1}-1))\) Using the product rule and simplifying, we find: \(P'(x) = \frac{50-2x}{(x+1)^2}\) Now, set P'(x) equal to zero and solve for x: \(\frac{50-2x}{(x+1)^2} = 0\) \(50-2x = 0\) \(x=25\)

Step 6: Find the printed area dimensions

Now that we have found the width x, we can find the length y by substituting x back into the expression for y in terms of x: \(y = \frac{50}{x+1} - 1\) \(y = \frac{50}{25+1} - 1\) \(y = 1\) So, the dimensions of the printed area that will result in the maximum printed area are width x=25 inches and length y=1 inch.

Key Concepts

Page Layout Design

Designing the layout of a page in a book involves more than just arranging text and images. A crucial part of page layout design is determining the optimal use of space so that the content is visually appealing and functional. For a book designer, such as in our example, achieving a balance between margins and print area is essential.
To break it down:

• The total page has specified margins of 1 inch on top and bottom, and 0.5 inch on each side.
• These margins are non-negotiable and help frame the text, making it easier for readers to focus on the content.
• The focus is on optimizing the printed area within these constraints, while the overall page area remains fixed at 50 square inches.

Understanding the structure imposed by fixed margins helps in planning the design to both aesthetic and practical standards. This ensures the content is not only readable but also visually well-arranged.

Mathematical Modeling

Mathematical modeling involves expressing real-world problems through mathematical expressions and equations. In our page layout design problem, mathematical modeling is crucial. It allows us to describe the relationship between the physical page dimensions and the printed area.

The mathematical model begins with defining variables:

• The width of the printed area, denoted as \( x \), and the length, \( y \).
• Using the known margins, the total width and height of the page are \((x + 1)\) and \((y + 1)\), respectively.
• These dimensions lead to the equation representing the total area of each page: \((x + 1)(y + 1) = 50\).

Through steps like substituting and rearranging equations, this model helps us explore potential dimensions based on our constraints.
This method provides a systematic way of approaching optimization problems, as it lets us translate conditions into solvable equations.

Calculus Solutions

Calculus plays a pivotal role in finding the dimensions that maximize the printed area. In these optimization problems, finding the derivative helps us understand how a function's rate of change influences outcomes.
Here's how calculus aids our problem:

• The printed area function, \( P(x) = x\left(\frac{50}{x+1} - 1\right) \), is our primary focus as it relates to our variables \( x \) and \( y \).
• By computing the derivative \( P'(x) \), we look for critical points where the rate of change (slope) is zero because these are potential maxima or minima.
• Setting \( P'(x) = 0 \) gives us \( x = 25 \), revealing that with a width of 25 inches, the length is 1 inch, offering us maximum printed area on the page.

This application of calculus not only optimizes space but also demystifies how even slight adjustments in dimensions can significantly impact utility and design.