If exactly 198 people sign up for a charter flight, Leisure World Travel Agency charges $308/person. However, if more than 198 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Hint: Let x denote the number of passengers above 198.
Find the revenue function R(x).
R(x) = ?
Determine how many passengers will result in a maximum revenue for the travel agency.
passengers
What is the maximum revenue?
$ ?
What would be the fare per passenger in this case?
? dollars per passenger
To find the revenue function R(x), we need to consider two scenarios: when the number of passengers is exactly 198, and when the number of passengers is greater than 198.
1. When exactly 198 people sign up for the charter flight:
In this case, Leisure World Travel Agency charges $308 per person for all 198 passengers. The total revenue generated by this scenario can be calculated as:
Revenue (R1) = $308 * 198 = $61,184
2. When more than 198 people sign up for the charter flight:
Let x denote the number of passengers above 198, so the total number of passengers becomes 198 + x. For each additional person above 198, the fare is reduced by $1. Hence, the fare per person can be expressed as $308 - $1x.
The total revenue generated by this scenario can be calculated as:
Revenue (R2) = ($308 - $1x) * (198 + x)
Now, we can derive the revenue function by combining the two scenarios:
R(x) = R1 when x = 0 (198 passengers) and R(x) = R2 when x > 0 (number of passengers greater than 198)
R(x) = $61,184 when x = 0
R(x) = ($308 - $1x) * (198 + x) when x > 0
To find the number of passengers that will result in maximum revenue for the travel agency, we need to find the value of x that maximizes the revenue function R(x).
To do this, we can calculate the derivative of the revenue function R(x) with respect to x and set it equal to zero:
dR/dx = 0
Solving this equation will give us the value of x that maximizes the revenue.
Once we find the value of x, we can substitute it back into the revenue function R(x) to get the maximum revenue.
Additionally, to find the fare per passenger in this case, we can substitute the value of x into the expression $308 - $1x from the revenue function R(x). This will give us the fare per passenger in dollars.