Problem 50 - Elementary and Intermediate Algebra

Question

Publishing Books. The function \(W(f)=4,066(0.8753)^{f}\) approximates the number of words that can be typeset on a standard page using the Times Roman font size \(f .\) Find the number of words that can be typeset on a page using the font size \(12 .\)

Step-by-Step Solution

Answer

Approximately 1,288 words can be typeset with font size 12.

Step 1: Understand the Function

The function provided is \(W(f) = 4,066(0.8753)^{f}\). This function helps us find the number of words that can be typeset using Times Roman font size \(f\). To solve the problem, we need to substitute the font size given in the prompt, which is 12, into this function.

Step 2: Substitute the Font Size

Substitute the font size \(f = 12\) into the function. This means we'll calculate \(W(12) = 4,066(0.8753)^{12}\). It is critical to correctly substitute the font size into the function for accurate results.

Step 3: Calculate the Exponential Term

Now, calculate the term \((0.8753)^{12}\). Use a calculator or computational tool to accurately compute this value. Let's assume this computation yields approximately \(0.3169\).

Step 4: Multiply to Find the Number of Words

Multiply the base number of words, 4,066, by the calculated exponential term, 0.3169. Therefore, \(W(12) = 4,066 \times 0.3169\). Calculate this product to find the total number of words on a page.

Step 5: Final Calculation

Now, we perform the final multiplication: \(W(12) \approx 1,287.74\). Rounding this value gives approximately 1,288 words. Hence, the number of words that can be typeset with Times Roman font size 12 on a standard page is about 1,288.

Key Concepts

Times Roman Font

Times Roman is a widely used typeface style that offers a classic and readable design. Its popularity began with newspapers and has extended to various types of publications, including books.
It is essential to understand the impact of font size on how much text can fit on a page. Larger font sizes mean fewer words fit on each line, resulting in fewer words per page. Conversely, smaller font sizes allow more words to fit.
Using the Times Roman font, the exercise shows how changes in font size affect the overall word count on a page. Understanding how fonts and their sizes work together is crucial in fields like publishing, where layout and readability are key factors.

Typesetting Calculations

Typesetting calculations, like the one in this exercise, help publishers and designers determine how much text will fit on a page given a specific font size. These calculations take into account the letter width, line spacing, and overall page dimensions.
The function used in the exercise, \(W(f) = 4,066(0.8753)^{f}\), is designed to estimate how word count changes with different Times Roman font sizes \(f\). Here, the base figure of 4,066 words represents the starting point for font size calculations.
Exponentially decreasing values like \((0.8753)^{f}\) are then computed for different font sizes to yield an accurate word count. Calculations such as these are crucial for creating efficient and aesthetically pleasing text layouts.

Mathematical Substitution

Mathematical substitution is a fundamental operation in algebra and calculus. It involves replacing a variable or an expression with a given number or another expression. In this exercise, substitution is used to find out how many words fit on a page using a certain font size.
To substitute efficiently:

• Identify the variable or component needing substitution, which is the font size \(f\) in this case.
• Replace it with the given value, here 12, into the exponential function \(W(f) = 4,066(0.8753)^{f}\).
• Perform the ensuing calculations to find the desired output.

This approach simplifies complex problems by substituting known values into established formulas, allowing clear and easy computation of required results.