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Chapter 1: Problem 6

Preparing a letter: You pay your secretary \(\$ 9.25\) per hour. A stamped envelope costs 50 cents, and paper costs 3 cents per page. a. How much does it cost to prepare and mail a 3 -page letter if your secretary spends 2 hours on typing and corrections? b. Use a formula to express the cost of preparing and mailing a letter as a function of the number of pages in the letter and the time it takes your secretary to type it. Identify the function and each of the variables you use, and state the units. c. Use the function you made in part b to find the cost of preparing and mailing a 2 -page letter that it takes your secretary 25 minutes to type. (Note: 25 minutes is \(\frac{25}{60}\) hour.)

Short Answer

Expert verified
3-page letter costs $19.09. Cost function: \( C(h, p) = 9.25h + 0.50 + 0.03p \). 2-page letter costs $4.41.

Step by step solution

01

Calculate Secretary's Cost for 2 Hours

To find the cost of your secretary's time spent on the letter, multiply the hourly rate by the number of hours:\[ \text{Secretary's Cost} = 9.25 \times 2 = 18.50 \text{ dollars} \]
02

Calculate Material Costs for a 3-Page Letter

This involves calculating the cost of the stamped envelope and the cost of paper:- **Envelope cost**: 0.50 dollars- **Paper cost**: 3 pages at 3 cents each: \[ \text{Paper Cost} = 3 \times 0.03 = 0.09 \text{ dollars} \]
03

Total Cost for a 3-Page Letter

To find the total cost, add up all the individual costs:\[ \text{Total Cost} = 18.50 + 0.50 + 0.09 = 19.09 \text{ dollars} \]
04

Define the Cost Function

Let \( C \) be the total cost, \( h \) be the hours the secretary spends, and \( p \) be the number of pages. The cost function is:\[ C(h, p) = 9.25h + 0.50 + 0.03p \]- **\( h \)**: hours (units in hours)- **\( p \)**: number of pages (units in pages)
05

Use Function to Calculate Cost for a 2-Page Letter

Convert the secretary's time into hours: \( \frac{25}{60} \approx 0.4167 \text{ hours} \). Substitute this time and the number of pages (2) into the function:\[ C(0.4167, 2) = 9.25 \times 0.4167 + 0.50 + 0.03 \times 2 \]\[ C = 3.85475 + 0.50 + 0.06 = 4.41475 \text{ dollars} \]
06

Round and Present Final Cost

Round the final cost to two decimal places:\[ C \approx 4.41 \text{ dollars} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variables Identification
Before solving any cost function problem, it is crucial to identify all relevant variables. This involves recognizing each element that contributes to the total cost. In our example, we have the following variables:
  • **Secretary's Time (h):** The time taken by your secretary to type and make corrections on the letter. The unit for this variable is hours.

  • **Number of Pages (p):** The number of pages that the letter contains. This affects the total cost of paper and, implicitly, the overall expense. The unit here is simply pages.

  • **Material Costs:** Though not directly variable, aspects like the cost of an envelope and the cost per page of paper also need to be considered as constants in the function.
Recognizing these variables helps in defining the function and makes it easier to calculate the entire cost accurately.
Formulating Functions
Once you've identified the variables involved, the next step is to formulate a cost function. A cost function is a mathematical expression that provides a total expense based on various input variables.In our scenario, the function captures:- **The secretary's time cost:** Calculated as the product of the hourly rate (\(9.25) and the number of hours worked (h).- **Fixed costs:** Elements like a stamped envelope (\)0.50) and the cost per page of paper ($0.03).With this in mind, the cost function can be expressed as:\[ C(h, p) = 9.25h + 0.50 + 0.03p \]Here, **C** is the total cost, **h** represents hours, and **p** represents the number of pages. Formulating the function in such a manner allows for easy computation of total costs under different conditions.
Step-by-step Calculation
With the cost function and variables well-defined, we can proceed with our calculations. Here's a structured approach:1. **Convert Variables Appropriately:** If input time is in minutes, convert it into hours to match the units used in the function. For instance, 25 minutes converts to hours by calculating \( \frac{25}{60} \approx 0.4167 \).2. **Substitute Values into the Function:** To find the cost of a 2-page letter where the secretary types for 25 minutes, substitute respective values into the function:\[ C(0.4167, 2) = 9.25 \times 0.4167 + 0.50 + 0.03 \times 2 \]3. **Perform Arithmetic Calculations:** Follow basic arithmetic steps to calculate: - Multiply the secretary's hourly rate by the hours worked: \(9.25 \times 0.4167 = 3.85475 \) - Add the fixed costs: Principal envelope (\(0.50) plus per page for two pages (\)0.06).4. **Total Cost Calculation:** Add these components to reach a grand total: \[ C = 3.85475 + 0.50 + 0.06 = 4.41475 \text{ dollars} \]5. **Round the Result for Practical Use:** It's usually practical to round the cost to two decimal places. Thus, \(C \approx 4.41 \text{ dollars}\). This step ensures the cost is presented in an easily understandable format.By following these precise steps, you ensure the calculation is accurate, helping avoid any financial oversight and making the outcomes clear.

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Most popular questions from this chapter

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