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Chapter 1: Problem 7

Preparing a letter, continued: This is a continuation of Exercise 6. You pay your secretary \(\$ 9.25\) per hour. A stamped envelope costs 50 cents, and regular stationery costs 3 cents per page, but fancy letterhead stationery costs 16 cents per page. Assume that a letter requires fancy letterhead stationery for the first page but that regular paper will suffice for the rest of the letter. a. How much does the stationery alone cost for a 3-page letter? b. How much does it cost to prepare and mail a 3 -page letter if your secretary spends 2 hours on typing and corrections? c. Use a formula to express the cost of the stationery alone for a letter as a function of the number of pages in the letter. Identify the function and each of the variables you use, and state the units. d. Use a formula to express the cost of preparing and mailing a letter as a function of thenumber 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. e. Use the function you made in part \(d\) to find the cost of preparing and mailing a 2 -page letter that it takes your secretary 25 minutes to type.

Short Answer

Expert verified
Stationery alone for a 3-page letter costs 22 cents. Preparing and mailing it costs $19.22. The formula for stationery cost is \(S(p) = 16 + 3\times(p - 1)\) cents. Preparing and mailing cost is \(C(p, t) = (16 + 3\times(p-1) + 50)/100 + 9.25 \times t\). A 2-page letter with 25 minutes prep costs $5.237.

Step by step solution

01

Calculate stationery cost for a 3-page letter

The first page uses fancy letterhead, costing 16 cents. Each additional page uses regular stationery, costing 3 cents per page. For a 3-page letter, one page costs 16 cents and two pages cost \(2 \times 3 = 6\) cents. So, total stationery cost is \(16 + 6 = 22\) cents.
02

Calculate total cost for preparing and mailing the 3-page letter

The stationery cost calculated in Step 1 is 22 cents. A stamped envelope costs 50 cents, totaling \(22 + 50 = 72\) cents. The secretary spends 2 hours with a wage of \(\$9.25\) per hour, totaling \(2 \times 9.25 = 18.50\). Thus, the total cost is \(18.50 + 0.72 = 19.22\) dollars.
03

Define function for stationery cost as a function of pages

Let \(S(p)\) represent the stationery cost for a letter of \(p\) pages. The formula is \(S(p) = 16 + 3\times(p - 1)\) where \(p\) is the total number of pages. The unit of \(S(p)\) is cents.
04

Define function for the total cost of preparing and mailing a letter

Let \(C(p, t)\) be the total cost, \(p\) pages, \(t\) hours. Formula: \(C(p, t) = (16 + 3\times(p-1) + 50)/100 + 9.25 \times t\). The first term calculates stationery and envelope cost (in dollars) and the second is secretarial cost (in dollars).
05

Calculate cost for a 2-page letter taking 25 minutes to type

A 2-page letter requires one fancy page and one regular page, thus \(S(2) = 16 + 3 \times 1 = 19\) cents. Total mailing cost: 69 cents. Secretary time is \(t = 25/60\approx 0.417\) hours. Total cost \(= 9.25 \times 0.417 + 0.69 = 4.547 + 0.69 = 5.237\) dollars.

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Key Concepts

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

Stationery Costs
Understanding stationery costs is crucial for managing expenses in any office environment. Stationery includes items like paper, pens, and envelopes needed for sending letters or documents.

In our exercise, stationery costs are broken down into two types: regular stationery and fancy letterhead stationery. Regular stationery costs 3 cents per page, whereas fancy letterhead, which is used for the first page of a formal letter, costs 16 cents per page. When calculating the total stationery cost for a letter, you need to consider the type and number of pages used.

For instance, in a 3-page letter, only the first page uses the expensive stationery, while the remaining pages use the cheaper option. Therefore, calculating stationery costs accurately can help in budgeting for routine office correspondence.
Letter Preparation Cost
Calculating the letter preparation cost involves understanding both stationery and labor expenses.

Besides the cost of the papers and envelopes, the time spent by the secretary on typing and correcting the letters contributes to the letter preparation cost. You pay the secretary \(\$9.25\) per hour for their work. Thus, assessing how long they spend on the task directly correlates with the cost.

For example, typing and preparing a 3-page letter may take 2 hours, resulting in a labor cost of \(2 \times 9.25 = 18.50\) dollars. Added to this is the cost of stationery and mailing (stamped envelope), leading to a comprehensive understanding of total preparation costs.
Algebraic Modeling
Algebraic modeling simplifies the calculation of costs through the creation of functions or formulas.

In our exercise, algebraic modeling allows us to express costs as functions of page numbers and time. The formulation of functions like \(S(p) = 16 + 3\times(p - 1)\) helps in understanding how costs accumulate based on the number of pages.
For overall preparation and mailing costs, the function \(C(p, t) = (16 + 3\times(p-1) + 50)/100 + 9.25 \times t\) ties the number of pages \(p\) and the time \(t\) spent typing into a comprehensive cost analysis equation. This method enhances accuracy in predictions and budgeting for office tasks.
Time Management in Office Tasks
Time management is crucial for efficiency and cost-effectiveness in office tasks.

In the exercise, the secretary's hourly wage and the time taken to type and prepare letters directly affect overall costs. By understanding and shortening the time required for routine tasks, offices can optimize productivity and reduce expenses.
For example, if typing a letter takes 25 minutes or approximately 0.417 hours, calculating the cost for this task becomes straightforward \(9.25 \times 0.417\). Time management strategies could involve utilizing templates, improving typing speed, or reducing errors, thus saving both time and money daily.

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Most popular questions from this chapter

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