It costs a bus company 225$ to run a bus on a ski trip, plus 30$ per passenger. The bus has a seating capacity of 22 passengers. The company charges 60$ per fare if the bus is full. For each empty seat, the company has to increase the ticket price by 60$. Explain how to determine maximum profit.
wait. you say that if there are 22 passengers, each ticket costs $60
but if there is one empty seat, each ticket costs $120?
and if there are 2 empty seats, each ticket costs $180?
I don't think so!
But assuming that's so, let's go on, and you can fix it as needed.
Total cost for x riders: 225+30x
total revenue: 60(23-x)*x
profit: 60x(23-x)-(30x+225) = -60x^2 + 1350x - 225
the maximum profit is at the vertex, which I'm sure you can find...
To determine the maximum profit, we need to consider the number of passengers that will maximize the profit for the bus company.
Step 1: Calculate the profit for each possible number of passengers.
- Let's assume x represents the number of passengers on the bus.
- The cost to run the bus is a fixed cost of $225.
- The cost per passenger is $30.
- The fare charged per fare is $60.
- For each empty seat, the ticket price increases by $60.
The profit can be calculated using the formula: Profit = Revenue - Cost.
Revenue can be calculated as: Revenue = Number of passengers * Fare.
The cost can be calculated as: Cost = Fixed cost + (Cost per passenger * Number of passengers).
Step 2: Determine the number of passengers that maximize the profit.
- Determine the revenue and cost for each possible number of passengers.
- Calculate the profit for each case (Profit = Revenue - Cost).
- Find the number of passengers that gives the highest profit.
Step 3: Calculate the maximum profit.
- Use the number of passengers that gives the highest profit.
- Calculate the revenue, cost, and profit using the formulas mentioned in Step 1.
By following these steps, we can determine the maximum profit for the bus company on a ski trip.
To determine the maximum profit, we need to find the optimal number of passengers that will maximize revenue and minimize costs for the bus company.
Here's how we can approach this:
1. Calculate the revenue: The revenue for a fully booked bus of 22 passengers can be calculated as the number of passengers multiplied by the fare price ($60): Revenue = 22 * $60.
2. Calculate the cost: The cost for running the bus trip is a fixed cost of $225, plus a variable cost of $30 per passenger. So, the cost can be calculated as: Cost = $225 + (Number of passengers * $30).
3. Calculate the profit: Profit is the difference between revenue and cost: Profit = Revenue - Cost.
4. Determine the maximum profit: To find the maximum profit, we need to determine the number of passengers that will yield the highest profit. We can do this by trying different numbers of passengers (from 0 to 22) and calculating the profit for each scenario.
- Start with 0 passengers and calculate the profit.
- Incrementally increase the number of passengers by 1 and calculate the profit each time.
- Stop when adding one more passenger results in a decrease in profit compared to the previous scenario.
5. Once we have determined the number of passengers that results in the highest profit, we can calculate the actual profit by substituting that number back into the profit equation.
6. Finally, we can compare this maximum profit to the scenario where the bus is full to see if it is more profitable to partially fill the bus and increase the fare.
By following these steps, we can optimize the number of passengers and ticket prices to maximize the profit for the bus company.