Problem 80
Question
A chartered bus company has the following price structure. A single bus ticket costs \(\$ 30 .\) For each additional ticket sold to a group of travelers, the price per ticket is reduced by \(\$ 0.50 .\) The reduced price applies to all the tickets sold to the group. (a) Calculate the total cost for one, two, and five tickets. (b) Using your calculations in part (a) as a guide, find a quadratic function that gives the total cost of the tickets. (c) How many tickets must be sold to maximize the revenue for the bus company?
Step-by-Step Solution
Verified Answer
The total cost for one, two, and five tickets will be $30, $59, and $140 respectively. The quadratic function that represents the total cost is \(T(n) = -0.5n^2 + 30n\). The maximum revenue is achieved when the bus company sells 30 tickets.
1Step 1: Calculate the Total Cost for One, Two, and Five Tickets
For one ticket, the cost is just the initial price, which is \$30. For two tickets, the bus company reduces the price per ticket by \$0.50, meaning each ticket costs \$29.50. Thus, the total cost for two tickets is \$29.50 * 2 = \$59. For five tickets, the price reduction now becomes \$0.50 * 4 = \$2, meaning each ticket now costs \$28. The total cost is therefore \$28 * 5 = \$140.
2Step 2: Finding a Quadratic Function for the Total Cost
The price per ticket decreases by \$0.50 for every ticket sold beyond the first, the total cost \(T\) as a function of the number of tickets \(n\) can be expressed as \(T(n) = n(30 - 0.5(n-1)) = 30n - 0.5n^2 + 0.5n\). Therefore, the quadratic function is \(T(n) = -0.5n^2 + 30n\)
3Step 3: Finding the Maximum Revenue for the Bus Company
To find the maximum of our quadratic function, we use the formula for the vertex of a parabola, \(h = -b/(2a)\), where \(a\) is the coefficient of \(n^2\) and \(b\) is the coefficient of \(n\). Substituting \(a = -0.5\) and \(b = 30\), we find that \(h = -30/(2*(-0.5)) = -30/-1 = 30\). Since \(h\) is the coordinate representing number of tickets, and number of tickets can only be an integer, the maximum revenue is achieved when the bus company sells 30 tickets.
Key Concepts
Ticket PricingRevenue MaximizationVertex of a Parabola
Ticket Pricing
Understanding ticket pricing is essential for finding the right balance between cost and sales. For each ticket sold in a large group, the price inclusively decreases, making group purchases more attractive. In our case:
This is a common strategy used by companies to boost sales. As more tickets are sold, customers benefit from the reduced price. This tactic effectively encourages more purchases, making the service more enticing for groups.
- The base ticket price is set at $30.
- An additional $0.50 reduction is applied per extra ticket bought after the first one.
This is a common strategy used by companies to boost sales. As more tickets are sold, customers benefit from the reduced price. This tactic effectively encourages more purchases, making the service more enticing for groups.
Revenue Maximization
Revenue maximization involves adjusting pricing based on quantity sold to maximize income. The key goal is to find the number of tickets sold at which revenue peaks.
To achieve maximum revenue, you need to understand the relationship between ticket numbers and price reductions, resulting in a total revenue formula. By manipulating the formula, you can predict the sales and price point that yield the highest income for a company.
This strategic approach ensures that all possible profits are considered when taking pricing decisions.
- Firstly, identify the quadratic function which models revenue. In this exercise, it's important to note that the revenue function will form a parabolic shape when plotted on a graph.
- The apex of this parabola, the vertex, will represent the maximum revenue achievable.
To achieve maximum revenue, you need to understand the relationship between ticket numbers and price reductions, resulting in a total revenue formula. By manipulating the formula, you can predict the sales and price point that yield the highest income for a company.
This strategic approach ensures that all possible profits are considered when taking pricing decisions.
Vertex of a Parabola
The vertex of a parabola is crucial in various real-world applications. In the context of ticket sales and pricing, finding the vertex pinpoints the ticket quantity that maximizes revenue.
For quadratic functions, the vertex (h, k) is found using the formula \[h = -\frac{b}{2a}\]where \(a\) and \(b\) are the coefficients from the standard quadratic equation \(ax^2 + bx + c\).
In our example:
By applying our coefficients to the vertex formula, the number of tickets for maximal revenue is calculated to be 30.
Understanding how to find and interpret the vertex enables optimal decision-making for maximizing revenue and forecasting potential sales.
For quadratic functions, the vertex (h, k) is found using the formula \[h = -\frac{b}{2a}\]where \(a\) and \(b\) are the coefficients from the standard quadratic equation \(ax^2 + bx + c\).
In our example:
- \(a = -0.5\), reflecting how each additional ticket further decreases total price per ticket.
- \(b = 30\), representing the original ticket price.
By applying our coefficients to the vertex formula, the number of tickets for maximal revenue is calculated to be 30.
Understanding how to find and interpret the vertex enables optimal decision-making for maximizing revenue and forecasting potential sales.
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