Problem 52 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

At the Central Perk coffeehouse in Manhattan, Rachel serves \(c\) cups of coffee and \(d\) desserts per hour. The coffee costs \(a\) dollars per cup, and the desserts cost \(b\) dollars each. She averages a tip of \(k\) cents per dollar of the customers' bills (excluding taxes). In addition, she makes a fixed wage of \(F\) dollars per hour. Consider \(c, d, a, b, k\), and \(F\) as constants. Express Rachel's earnings as a function of \(h\), the number of hours she works. (In actuality, Rachel's earnings are not a function of the hours she puts in. Other considerations complicate the situation. For instance, business is slow at certain times of the day, and some customers tip more generously than others. Nevertheless, by using the information provided, we can make a mathematical model of the situation that gives us a reasonably accurate picture.)

Step-by-Step Solution

Answer

The earnings function is given as \(E(h) = h*(F + 0.01*k*(c*a + d*b))\) USD.

Step 1: Calculate total revenue from coffee and desserts

First, the total revenue from selling items per hour has to be calculated. Therefore, the revenue is the product of the quantity sold and the price for each item. Hence, for \(c\) cups of coffee at \(a\) dollars per cup and \(d\) desserts at \(b\) dollars each, the revenue earned per hour in dollar terms is given by \(c*a + d*b\). This needs to be done since Rachel earns tips on the basis of the revenue generated from the sales.

Step 2: Calculate Tips

Next, the tips earned need to be computed. Tips are a function of the total bill. Therefore, Rachel earns \(k\) cents per dollar of the revenue. Hence, tips are given by \(0.01*k*(c*a + d*b)\). It's important to multiply the result by 0.01 because \(k\) is given in cents, and we need to express it in dollars to add it to Rachel's total earnings.

Step 3: Calculate total earnings

Finally, Rachel's total earnings per hour are obtained by adding up the fixed wage and the tips Rachel has made, i.e., \(F + 0.01*k*(c*a + d*b)\).

Step 4: Calculate Rachel's earnings for \(h\) hours

The quantity of interest is Rachel's earnings as a function of \(h\), i.e., the number of hours she works. Hence, to calculate the earnings for \(h\) hours, simply multiply the earnings per hour derived in step 3 by \(h\). Therefore, function \(E(h) = h*(F + 0.01*k*(c*a + d*b))\).

Key Concepts

Revenue Calculation

Revenue calculation is a key part of understanding how much money a business or an individual makes from selling goods or services. In Rachel's case at Central Perk, she sells two items: coffee and desserts. Here's how we break down the revenue calculation:

• Coffee sales: Rachel sells a specific number of cups of coffee every hour, denoted by c. Each cup costs a dollars. Thus, the hourly revenue from coffee is the product of cups sold and the price per cup, or \( c \times a \).
• Dessert sales: Similarly, she sells d desserts per hour, each costing b dollars. The revenue for desserts is calculated as \( d \times b \).
• Total revenue: To find the total revenue from both products in any given hour, we sum these amounts. That's \( c \times a + d \times b \).

This calculation gives us an estimate of how much money Central Perk makes from Rachel's sales per hour. This is crucial as the tips she receives are based on this revenue.

Tips Calculation

Tips are an essential part of earnings for service industry workers, like Rachel at Central Perk. Calculating tips involves a simple mathematical step, and here's why it's important:

• Understanding tips: Rachel earns tips based on the total revenue she generates through sales. However, tips are not a direct percentage of the revenue; instead, they are calculated as a fixed amount per dollar.
• Tips in cents: Rachel averages a tip of k cents for every dollar spent by the customers. To convert this into dollars (since Rachel's earnings will be calculated in dollars), we multiply by 0.01. Thus, her tips are calculated as \( 0.01 \times k \times (c \times a + d \times b) \).

This step helps us integrate tips into Rachel's total earnings. Understanding tips this way helps in accurately modeling earnings for Rachel and other similar scenarios.

Earnings Function

An earnings function is a mathematical model that helps us understand Rachel's total earnings over time. It's crucial to calculate this correctly to have a precise picture of her income. Here's how we build the earnings function:

• Fixed Wage: Rachel earn a fixed wage of F dollars per hour. This means no matter how many coffees or desserts she sells, she gets this amount every hour.
• Total hourly earnings: Every hour, her earnings are a combination of her fixed wage and the tips she makes from selling coffee and desserts. We calculated earlier as \( F + 0.01 \times k \times (c \times a + d \times b) \).
• Earnings over several hours: To find out how much Rachel earns over a period h hours, simply multiply her hourly earnings by the number of hours. Hence, the function for Rachel's earnings can be expressed as \( E(h) = h \times (F + 0.01 \times k \times (c \times a + d \times b)) \).

This function helps simplify and predict earnings based on the time Rachel works, encompassing all elements affecting her total income.