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Chapter 0: Problem 27

A book designer 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. \(^{2}\). Find a function in the variable \(x\), giving the area of the printed page (see the figure). What is the domain of the function?

Short Answer

Expert verified
The function for the area of the printed portion of the page is: \(A(x) = (x-1)\left(\frac{50}{x} - 2\right)\), and the domain of the function is all real numbers \(x > 25\).

Step by step solution

01

Define variables

Let the length of the page be \(x\) inches. Then the width of the page will be \((50/x)\) inches.
02

Determine the dimensions of the printed area

Now, we need to subtract the margins from the length and width to find the dimensions of the printed area. Since there are 1-inch margins at the top and bottom, and \(\frac{1}{2}\)-inch margins on the sides, the dimensions will be: Length of printed area: \(x-2(\frac{1}{2})=x-1\) Width of printed area: \((50/x)-2(1) = (50/x)-2\)
03

Write a function for the area of the printed portion

The area of the printed portion can be found by multiplying the dimensions of the printed area: Area printed portion (A) = (Length of printed area) * (Width of printed area) Therefore, \(A(x) = (x-1) \left(\frac{50}{x} - 2\right)\)
04

Find the domain of the function

The domain of a function represents the valid values of \(x\) for which the function is defined. In this case, \(x\) must be a positive value, since lengths cannot be negative, and \(x \neq 0\), since we cannot divide by zero. Furthermore, the printed area must have a positive area, so both length and width of printed area must be positive, i.e., \(x - 1 > 0\) and \(\frac{50}{x} - 2 > 0\). Solving these inequalities, we get: 1. \(x > 1\) 2. \(x > \frac{50}{2} = 25\) So, the domain for the function \(A(x)\) is all real numbers \(x > 25\).
05

Final Result

The function for the area of the printed portion of the page is: \(A(x) = (x-1)\left(\frac{50}{x} - 2\right)\) And the domain of the function is all real numbers \(x > 25\).

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