Problem 18 - Calculus for Biology and Medicine

Question

Ticket Price Optimization Dalmatian Airlines also flies a daily flight from Los Angeles to Sacramento. Currently they sell each ticket for \(\$ 100\), and on average 200 people take the flight, so their revenue per flight is 200 tickets \(\times \$ 100 /\) ticket \(=\$ 20,000 .\) They are interested in seeing whether they can increase their revenue by changing the price of a ticket. Based on market research they discover that for every \(\$ 2\) increase in ticket price, one fewer person will buy a ticket. Similarly for every \(\$ 2\) decrease in ticket price, one more person will buy a ticket. (a) What ticket price would maximize Dalmatian Airlines' revenue? (Hint: Denote the number of extra people flying on the route due to a price change by \(x\), and the cost of a ticket by \(\$ 100-2 x .\) Then explain why the revenue to be maximized is \(R(x)=(100-2 x)(200+x)\). You should also explain what the domain of this function is.) (b) The plane can seat a maximum of 250 people. How does this information change the domain of \(R(x) ?\) Does this constraint affect your answer to part (a)?

Step-by-Step Solution

Answer

Keep ticket price at $100 (200 passengers) for maximum revenue within constraints.

Step 1: Define Revenue Function

To maximize revenue, define the function where revenue is influenced by ticket price and number of passengers. Let the change in number of passengers be denoted by \( x \). If ticket price changes by \(-2x\), the passenger change is \( x \). The revenue function, using the hint, is given by:\[R(x) = (100 - 2x)(200 + x)\]This represents the product of ticket price and number of passengers.

Step 2: Expanding the Revenue Formula

Expand the revenue function \( R(x) = (100-2x)(200+x) \) to standard polynomial form:\[R(x) = 100(200) + 100x - 2x(200) - 2x^2 \R(x) = 20000 + 100x - 400x - 2x^2 \R(x) = 20000 - 300x - 2x^2\]

Step 3: Identify Constraints for \(x\)

The problem states each \\(2 change in price results in \(x\) tickets sold more or less. Initial passengers are 200. Domain analysis gives:- Minimum price: \\)0 resulting in \(x = 50\) (250 passengers at most)- Maximum price: All passengers leave if price too high, \(x \leq 200\)Thus, the domain for \(x\) is \(-200 \leq x \leq 50\).

Step 4: Determine Maximum Revenue using Quadratic Formula

To find the maximum revenue, complete the square for the quadratic equation or use vertex formula. For quadratic functions \( ax^2 + bx + c \), vertex occurs at:\[x = -\frac{b}{2a} = -\frac{-300}{2 \cdot -2} = 75\]Since 75 is outside domain, max decisions are checked within domain.

Step 5: Calculate Revenue within the Valid Domain

Structure domains, rather than whole real line check constrained maximum:For \(x = 50\):\[R(50) = (100 - 2 \cdot 50)(200 + 50) = 0\cdot 250 = 0\]For \(x = 0\) (original state):\[R(0) = 20000\]Conclude \(x = 50\) (\$0 price) not feasible, thus stick to orig at 200 passengers.

Step 6: Evaluate Plane Seat Limitation

Given maximum seats plane holds is 250 passengers. This constraint defines new passenger constraints:- The largest \( x \) such that \( 200 + x \leq 250 \)- For \(x \leq 50\)Already considered with previous calculations. Conclusion confirmed, orig no seat impact for \( x = 0 \).

Key Concepts

Quadratic Functions

Quadratic functions play a crucial role in revenue optimization problems, particularly when analyzing situations where the change in price affects the number of units sold. In this context, the revenue function is modeled as a quadratic function. Let's first understand what this means.

A quadratic function is generally expressed in the form:

• \(ax^2 + bx + c\)

For Dalmatian Airlines, the given revenue function is:

• \(R(x) = (100 - 2x)(200 + x)\)

To better analyze it, we expand the function into standard polynomial form, which gives:

• \(R(x) = 20000 - 300x - 2x^2\)

Breaking it down, this represents how changes in ticket price (\(-2x\)) and passengers (\(x\)) affect the revenue. The term \(-2x^2\) suggests the revenue would initially increase and then eventually decrease, forming a parabola shape that opens downward. This parabolic curve helps us find the maximum or minimum points of the function. This understanding is key to optimizing revenue in a structured way.

Constraint Analysis

Constraint analysis is essential to ensure our solutions are realistic and applicable to the situation.

In this problem, constraints play a pivotal role in determining feasible solutions. Dalmatian Airlines faces multiple constraints:

• Price Change: For every \( \$2 \) change in price, the number of passengers changes by \( x \).
• Seating Capacity: The plane can hold a maximum of 250 passengers.

Let's delve deeper into these.Firstly, the pricing adjustment constraint means that if the price decreases, more passengers travel, and vice versa. Mathematically, if the price were to decrease, more passengers would potentially fly. However, if we increase the price, fewer passengers will be interested.The airplane's seating capacity further constrains the number of passengers. Therefore, only outcomes where the total number of passengers is \( \leq 250 \) are valid. Hence, this affects the domain and potential price adjustments we can implement. By adhering to constraints, we ensure realistic and practical solutions are derived.

Domain of a Function

The domain of a function in a business context refers to all the possible input values ("x") that allow the function to operate effectively.

For the Dalmatian Airlines scenario, determining the valid domain for the revenue function, \( R(x) = (100 - 2x)(200 + x) \), is critical as it identifies all permissible values that result in valid revenue.
When analyzing this domain:

• Minimum Price Consideration: The lowest possible ticket price is \(0\), which translates to \(x = 50\) (because \(100-2x = 0\), thus \(2x = 100\)). This corresponds to 250 passengers onboard the aircraft.
• Maximum Price Consideration: In a scenario where ticket prices are excessively high and no passengers purchase them, with minimum number of passengers, theoretically, \(x\) should not exceed \(200\), but practically, the seating constraint limits \(x \leq 50\).

In this optimization problem, considering these constraints, the domain of function \( R(x) \) is limited to \(-200 \leq x \leq 50\). This established domain ensures no unrealistic results are computed, making sure all derived solutions are feasible within provided situational boundaries.