Problem 53 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

I work an \(h\) -hour day. I spend \(1 / w\) of these \(h\) hours on the road and the remainder in consultation. I receive \(A\) dollars per hour as a consultant. I receive no money when I'm on the road. In fact, each day I pay \(G\) dollars in gas and tolls and I estimate that each day costs \(C\) cents in wear and tear on the car. I have no other expenses. Express my daily profit as a function of \(h\), the number of hours I work. \((A, w, G\), and \(C\) are all constants.) (Note: If \(w=5\), I spend \(1 / 5\) of my workday on the road and the rest of my workday in consultation.)

Step-by-Step Solution

Answer

The daily profit as a function of \( h \) is represented by the equation \( P(h) = A \times (h - (h / w)) - (G + C / 100) \)

Step 1: Determine the time spent on the road

To know the hours spent on the road, and thus determine the hours spent as a consultant, we'll have to multiply the total hours by the fraction of time spent on the road. This amounts to \( h \times (1 / w) \).

Step 2: Determine the time spent in consultation

To get the time spent in consultation, subtract the time spent on the road from the total hours. This equals \( h - (h / w) \).

Step 3: Calculate earnings from consultation

Given that for every hour in consultation, you earn A dollars, the total earnings from consultation is, therefore, \( A \times (h - (h / w)) \).

Step 4: Compute the cost in dollars

Since costs are given in different units (dollars for gas and tolls, cents for wear and tear), it's necessary to convert them into a common unit. Convert \( C \) cents into dollars by dividing by 100. Total cost is \( G + C / 100 \).

Step 5: Subtract the cost from the earnings to get the daily profit

In the end, subtract the total cost from the total earnings in consultation to find daily profit. Therefore, profit is equal to \( A \times (h - (h / w)) - (G + C / 100) \).

Key Concepts

Consultation Hours

When calculating consultation hours, we first need to determine how much of the day you spend on the road. If you work for a total of \(h\) hours each day and a fraction \(1/w\) of these hours is spent traveling, then the remaining hours are devoted to consultation.

To find out the consultation hours:

• Calculate the hours spent on the road: \(h \times (1/w)\)
• Subtract road hours from total hours: \(h - (h/w)\)

This equation represents the time you actually spend working as a consultant with your clients, providing services or advice.

Understanding this helps in determining how many hours are billable, directly impacting your earnings.

Travel Expenses

Travel expenses can quickly add up and must be accounted for in any profit calculation. For each day you work, you incur certain costs due to traveling, specifically gas and tolls. These are fixed expenses you pay daily, regardless of how much consulting you perform.

Two main components make up these travel expenses:

• Gas and tolls: Represented by the constant \(G\) dollars per day.
• Wear and tear on the car: Estimated at \(C\) cents per day.

To include wear and tear as part of your cost calculations, convert \(C\) cents into dollars by dividing by 100. This gives you the complete daily travel expenses: \(G + C/100\).

Knowing your travel costs is essential for accurately assessing the profitability of your consultation work.

Consultant Earnings

Consultant earnings are directly tied to the number of hours you work in consultation. As a consultant, you earn a specified rate, \(A\) dollars, for each hour spent advising clients. To calculate your daily earnings:

• Identify total consultation hours: \(h - (h/w)\)
• Multiply consultation hours by your hourly rate: \(A \times (h - (h/w))\)

This calculation shows how much money you make before deducting any expenses, aligning your daily efforts worked with financial rewards. Balancing consultation time and travel time effectively maximizes your earnings.

Cost Calculations

Understanding cost calculations is vital in determining your net profit at the end of each day. You calculate profits by comparing your total earnings to your total expenses.

• Total earnings come from your consulting hours: \(A \times (h - (h/w))\)
• Total expenses include both your travel costs and wear and tear costs: \(G + C/100\)

Subtract your total expenses from your earnings to find daily profit: \[ A \times (h - (h/w)) - (G + C/100) \]
This formula simplifies your financial summary and provides a clear picture of how much you truly gain after covering your work-related expenses. Understanding this helps in making informed decisions about your business activities.