Problem 81 - College Algebra and Calculus: An Applied Approach

Question

For groups of 80 or more people, a charter bus company determines the rate per person (in dollars) according to the formula Rate \(=8-0.05(n-80) \quad n \geq 80\) where \(n\) is the number of people in the group. (a) Write the total revenue \(R\) for the bus company as a function of \(n\). (b) Complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline n & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\\\\hline R & & & & & & & \\ \hline\end{array}$$ (c) Write a paragraph analyzing the data in the table.

Step-by-Step Solution

Answer

The total revenue \(R\) for the bus company is given by the function \(R = n \times (8 - 0.05(n - 80))\). From the analysis of the filled table, it is noticeable that the total revenue increases with the number of people until it hits a maximum, and then starts decreasing as the number of people continues to increase.

Step 1: Derive the Function for the Total Revenue

From the given problem, rate = \(8 - 0.05(n - 80)\) where \(n\) is the number of people which is greater than or equal to 80. Total revenue (\(R\)) would then be derived from the product of the price per person (Rate) and the number of people (\(n\)). Hence, we find \(R = n \times \text{Rate}\) as the equation. Substitute the Rate from above to get the final equation for \(R\), \(R = n \times (8 - 0.05(n - 80))\).

Step 2: Complete the Table

Substitute each value of \(n\) from the table into the equation \(R = n \times (8 - 0.05(n - 80))\). Compute the result each time to find the corresponding \(R\) value. Fill in the table as follows: \n \n\[\begin{array}{|l|l|l|l|l|l|l|l|}\hline n & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\\hline R & 765 & 800 & 795 & 750 & 665 & 540 & 375 \ \hline\end{array}\]

Step 3: Analyze the Data in the Table

From the filled table, as the number of people in the group increase, the total revenue first increases, reaches a maximum, and then reduces. This indicates that having people more than a certain number will rather decrease the company's revenue instead of adding up to it.

Key Concepts

Rate Function

In algebra, a rate function describes how one quantity changes with respect to another. In the context of the charter bus company, the rate function represents the price per person, which varies depending on the number of people in the group (). The formula given, Rate = 8 - 0.05(n - 80) for ≥ 80, is an explicit function where the rate per person decreases by 5 cents for each person over 80.

Understanding the rate function is crucial for predicting how changing the number of passengers affects the individual price. For example, if a group has 90 people, the rate per person can be calculated by substituting 90 for in the rate formula, resulting in a rate of 7.50 dollars per person. It's necessary to decipher the elements of this function, such as the slope (-0.05), which indicates the rate of decrease in the price per additional person. As the number of passengers increases, the price per person continues to lower, which ultimately impacts the total revenue. This pricing strategy can encourage more extensive group bookings by offering lower rates for larger groups.

Total Revenue Calculation

Total revenue calculation is a fundamental concept in business and economics. It is determined by multiplying the price of a service or product by the quantity sold or number of users. For the charter bus company, the total revenue () is a function of the number of people in the group and the rate per person. The equation derived from the rate function is = n × (8 - 0.05(n - 80)).

To apply this, simply plug in the values of from the table into the total revenue equation. If there are 100 people in the group, the calculation would be = 100 × (8 - 0.05(100 - 80)), yielding a total revenue of 800 dollars. As seen in the filled table, the revenue changes with each increment of the group size. This kind of calculation helps the company project its earnings according to varying group sizes. A pattern or trend can often be observed, showing the optimal number of passengers for maximizing revenue, which is valuable for making strategic pricing decisions.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information, informing conclusions, and supporting decision-making. Analyzing the completed table for the charter bus company's revenue, a narrative emerges. Initially, as the number of passengers increases from 80 to 100, revenue grows. This is due to the additional passengers bringing in more money even though the rate per person is decreasing.

However, past a certain point, the total revenue starts to decline. This is because the rate reduction per additional person begins to outweigh the benefit of having more passengers. To retain profitability, the bus company must understand this balance. They may, for instance, determine an optimal number of passengers that maximizes revenue. Insights from this data can inform marketing campaigns, pricing strategies, and capacity planning, illustrating the enormous value of data analysis in a business setting.