philippine airlines transport around 800 passengers a month between manila and bacolod. a round trip ticket is 5,000.00 pesos. the company executives estimate that for each 300 pesos increase in the ticket price, 30 passengers would be lost to the competition. what ticket price would maximize income for the company? how much will be the revenue?
let the number of 300 pesos increases by n
after the change,
the cost of a ticket = 5000 + 300n
number of passengers = 800 - 30n
revenue = R = (5000+300n)(800-30n)
expand to get a quadratic function.
Find the vertex of that quadratic using your favourite method and you are almost there.
tnx
To determine the ticket price that would maximize the income for Philippine Airlines (PAL), we need to consider the relationship between ticket price, number of passengers, and revenue.
Let's break down the steps:
Step 1: Calculate the baseline revenue:
The current ticket price is 5,000 pesos, and PAL transports 800 passengers per month. So, the baseline revenue can be calculated as:
Baseline Revenue = Ticket Price * Number of Passengers
Baseline Revenue = 5,000 pesos * 800 passengers
Step 2: Determine the relationship between ticket price and lost passengers:
The executives estimate that for every 300 pesos increase in ticket price, 30 passengers would be lost to the competition.
Step 3: Calculate the number of passengers lost for each ticket price increase:
To calculate the number of passengers lost for various ticket price increases, we divide the price increase by 300 pesos and then multiply by 30 passengers:
Number of Passengers Lost = (Price Increase / 300) * 30
Step 4: Calculate the remaining number of passengers after each ticket price increase:
Number of Passengers Remaining = Initial Number of Passengers - Number of Passengers Lost
Step 5: Calculate the revenue for each ticket price:
Revenue = Ticket Price * Number of Passengers Remaining
Step 6: Find the ticket price that maximizes the revenue:
By calculating the revenue for different ticket prices, we can find the price that results in the highest revenue.
Now, let's calculate the revenue and find the ticket price that maximizes PAL's income:
Baseline Revenue = 5,000 pesos * 800 passengers
Baseline Revenue = 4,000,000 pesos
Let's calculate the revenue for different ticket prices, starting with a 300 pesos increase:
Price Increase: 300 pesos
Number of Passengers Lost = (300 pesos / 300 pesos) * 30 passengers
Number of Passengers Lost = 30 passengers
Number of Passengers Remaining = 800 passengers - 30 passengers
Number of Passengers Remaining = 770 passengers
Revenue = (5,000 pesos + 300 pesos) * 770 passengers
Revenue = 4,540,000 pesos
Continuing the calculation for other ticket prices, we find:
Ticket Price: 5,300 pesos
Number of Passengers Lost = (600 pesos / 300 pesos) * 30 passengers
Number of Passengers Lost = 60 passengers
Number of Passengers Remaining = 800 passengers - 60 passengers
Number of Passengers Remaining = 740 passengers
Revenue = (5,000 pesos + 600 pesos) * 740 passengers
Revenue = 4,444,000 pesos
Ticket Price: 5,600 pesos
Number of Passengers Lost = (900 pesos / 300 pesos) * 30 passengers
Number of Passengers Lost = 90 passengers
Number of Passengers Remaining = 800 passengers - 90 passengers
Number of Passengers Remaining = 710 passengers
Revenue = (5,000 pesos + 900 pesos) * 710 passengers
Revenue = 4,490,000 pesos
By continuing this calculation, we can find the ticket price that maximizes the revenue for PAL. However, without the precise values for the price increases and passenger losses, we cannot provide the exact ticket price or revenue amount.
To determine the ticket price that would maximize income for Philippine Airlines (PAL), we can use a simple optimization approach. Let's start by organizing the given information:
- Number of passengers transported per month: 800
- Price per round trip ticket: 5,000.00 pesos
- Expected passenger loss for each 300 pesos increase in ticket price: 30 passengers
Now, let's break down the problem step by step:
1. Calculate the initial revenue per month at the current ticket price.
Revenue = Number of passengers * Ticket price
Initial Revenue = 800 passengers * 5,000.00 pesos = 4,000,000.00 pesos
2. Determine the relationship between ticket price and passengers lost to the competition.
The executives estimate a passenger loss of 30 for each 300 pesos increase in ticket price. This can be expressed as:
Passengers Lost = (Increase in ticket price / 300 pesos) * 30 passengers
Since we want to maximize income, we need to minimize the number of lost passengers.
3. Calculate the number of passengers remaining at different ticket price points.
Remaining Passengers = Total Passengers - Passengers Lost
Remaining Passengers = 800 - [(Increase in ticket price / 300 pesos) * 30 passengers]
4. Calculate the revenue at different ticket price points.
Revenue = Number of remaining passengers * Ticket price
Revenue = [(800 - [(Increase in ticket price / 300 pesos) * 30 passengers])] * Ticket price
5. Determine the ticket price that maximizes revenue.
To find the optimal ticket price, we need to differentiate the revenue equation with respect to the increase in ticket price and set it to zero.
dRevenue/dIncrease in ticket price = 0
Solve this equation to find the ticket price that maximizes revenue.
Let's perform the calculations:
dRevenue/dIncrease in ticket price = d([(800 - [(Increase in ticket price / 300 pesos) * 30 passengers])] * Ticket price) / dIncrease in ticket price
= [800 - (0.1 * (30/300) * (800 - 30 * Ticket price / 300))] * Ticket price
Setting this equation to zero, we get:
800 - (0.1 * (30/300) * (800 - 30 * Ticket price / 300)) = 0
Simplifying the equation:
800 - (800 - 3 * Ticket price) = 0
3 * Ticket price = 800
Ticket price = 800 / 3 = 266.67 pesos
6. Calculate the revenue at the ticket price that maximizes income.
Using the optimal ticket price, we can calculate the final revenue:
Revenue = Number of remaining passengers * Ticket price
Revenue = [(800 - [(266.67 / 300) * 30])] * 266.67 pesos
Now you have all the information to answer the two questions:
1. The ticket price that would maximize income for PAL is 266.67 pesos.
2. Calculate the revenue using the optimal ticket price:
Revenue = [(800 - [(266.67 / 300) * 30])] * 266.67 pesos