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A charter bus company provides transportation services for groups of people. Whether it's a field trip, corporate outing, or a group vacation, these companies offer vehicles that can accommodate a large number of passengers.
In our scenario, the bus company aims to serve groups of 80 or more people. For practical purposes, such as budgeting and resource allocation, the company needs a reliable method to calculate costs and revenues. This is where rate and revenue functions come into play.
The company uses a specific formula to determine the price per person based on group size. This ensures both affordability for the customer and profitability for the company.

The rate per person is crucial for understanding how much each passenger will pay. In this problem, the rate is determined by a formula that takes the number of people into account.
The formula given is:

• Rate = 8 - 0.05(n - 80)

Here, 'n' represents the number of passengers. What this means is that the base rate starts at $8, but this decreases slightly as more people join the trip.
The decrease in rate encourages larger groups, offering cost savings per person as the group size increases. However, it's important for the company to monitor this, as the rate drop influences total revenue.

Revenue calculation is key to understanding a business's financial health. It's the total money collected from selling services or products, which in our case, is the bus trip service.
The formula we use for revenue comes from multiplying the rate by the number of passengers, or:

• Revenue, R(n) = Rate × n

By substituting the rate formula in, we get:

• R(n) = (8 - 0.05(n - 80))n

This equation captures how revenue is affected by both the number of passengers and the rate per person. Initially, increasing 'n' appears to increase revenue, but because the rate decreases slightly with each additional person, there’s a point where revenue plateaus and eventually declines.

Simplifying complex functions makes them easier to work with and understand. It helps to strip away unnecessary complexity, providing a clearer picture of what's happening.
To simplify the revenue function, we start with:

• R(n) = (8n - 0.05n^2 + 4n)

Combining like terms gives us:

• R(n) = -0.05n^2 + 12n

This simplified form of the revenue function highlights the quadratic relationship between revenue and the number of passengers.
It's clear now that as 'n' increases, there's a certain point where increasing 'n' leads to decreased revenue due to the negative coefficient of the quadratic term.
This simplification is vital for understanding and predicting revenue behavior as it makes it easier for the company to create strategies based on maximum and optimal passenger numbers.