Creating Functions as Models and Explaining Limitations of Models

In this lesson, after reviewing some concepts that were covered in the previous lesson, you will explore an example of how we can create a new function from existing functions. You will also consider how to explain limitations of a model in context. The problem is connected to the context of ticket sales for a rodeo and how we might find the ticket price that will maximize the income from the sale of tickets.

Let’s review your work from the previous lesson.

A. Using Polynomials to Solve Problems

Recall that in the previous lesson, we explored some examples of polynomial models in various contexts.

Worked Example
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We will revisit the problems of building an animal pen in the rodeo example from the last lesson's assessment.

Suppose that instead of having 100 feet of fencing for the animal pen, the rodeo manager only has 75 feet of fencing for the pen that will border the building. Then the dimensions of the pen are as shown below, with the width of the pen being 'x' and the length of the pen being 75 - 2x.

Screenshot 2024-11-07 at 6.55.35 PM.png

Part A: Create a function for the area of the pen.

Part B: Find the dimensions of the pen of maximum area

Solution:

Part A: Create a function for the area of the pen.

Since the area of a rectangle is width times length, the area function is :

$LaTeX: A\left(x\right)=x\left(75-2x\right)$

Part B: Find the dimensions of the pen of maximum area.

To find the dimensions of the pen with maximum area, graph the function using Desmos and click on the maximum point. Below is a graph of the area function with the maximum point shown.

Screenshot 2024-11-07 at 7.02.23 PM.png

We can see the maximum area is achieved with the width of the pen is 18.75 feet. To find the length of the pen, we need to calculate 75 – 2(18.75) = 37.5 feet. The dimensions of the pen of maximum area are 18.75 feet by 37.5 feet

Self Check:

Using the model for the area function in the Worked Example, explain why we must limit the domain of the function in the specific real-world context. And give the domain of the model in context.

Solution:

The function represents the area of the animal pen and the area must always be greater than 0. Using the Desmos graph, we can see that the parabola is greater than zero for x in the interval 0 and 37.5. The domain of the function in context is LaTeX: 0<x<37.5

Instructional Content:

Part I. Creating New Functions from Existing Functions

In this next example, we will consider some data connected to the price of multi-event ticket sales for rodeo fundraising and we will build a function to find the price of the multi-event ticket that would yield a maximum for the revenue generated from the sale of the tickets.

Worked Example
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Worked Example:

Need some disclaimer that this is not real data, But simulates the situation.

Student fundraising organization at NMHU wants to raise money to support the rodeo at the university by selling a ticket or pass that will allow the holder to attend various events at the university throughout the fall semester. The student organizers are trying to decide what price to charge for the ticket. Some students might argue for setting the price low, believing that a low price would bring a large response.

Others may want to charge a higher price, so that even if not many tickets are sold, they

would still make money. The student organizers decided to ask the families of the students what they would be willing to pay for a multi-events ticket or pass. They assume the families want the sale to be a success and will give them accurate information. A survey is sent out to families of students in the school asking, “What is the most you would be willing to pay for a multi-vent ticket good for one semester during the school year?”.

The data in the table below represents the results of that survey.

Screenshot 2024-11-07 at 7.19.04 PM.png

Below is graph of this data with the price for the multi-event ticket on the x-axis and the number of families willing to pay that price on the y-axis.

Screenshot 2024-11-07 at 7.21.25 PM.png

Screenshot 2024-11-07 at 7.22.08 PM.png

Question: Why do you think the data exhibits a decreasing trend?

Solutions:

The data is decreasing because as the price of the ticket increases, the number of families willing to pay that price decreases.

Because the data appears to have a linear relationship, we will find a linear model using Desmos.

A possible linear model for this data is

Number of Families = -5.08(ticket price) + 996

If we let x = ticket price. We can write the function as follows:

$LaTeX: F\left(x\right)=-5.08x+996.85$

Below is a scatter plot of the data with the linear model shown, We can see that the linear model provides a good representation of the data.

Screenshot 2024-11-07 at 7.35.13 PM.png

Self-Check

INSTRUCTION: Explain why we should limit the domain of the model in this context, And give the domain of the linear model in context.

Solution:

The x-coordinate of the data represents the price of a multi-event ticket and that number must always be greater than 0. Using the Desmos graph, we can see that the line intersects the x-axis and we can find that point of intersection by setting the function equal to 0 and solving for x.

LaTeX: 0=-5.08x+996.85

LaTeX: 5.08x=996.85

$LaTeX: x=\frac{996.85}{5.08}$

$LaTeX: x\approx196.27$

The domain of the function in context is LaTeX: 0<x<196.27

Self-Check

INSTRUCTION: Graph the model for the revenue function. Use the function to find the price of the ticket that will yield a maximum revenue. Round your answer to a value that is reasonable in terms of the price of the multi-event ticket.

Solution:

Screenshot 2024-11-07 at 8.04.49 PM.png

The maximum of the revenue graph occurs when x = 98.115. Since this will be the price of the multi-event ticket, the students may choose to charge $100 or $95 because people are used to paying for tickets in multiples of $5 increments.

We have seen in our previous examples that an important part of mathematical modeling is explaining the limitations of your model. The limitations can be related to the domain and range of the model specific to the context.

Creating Functions as Models and Explaining Limitations of Models

Worked Example

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Worked Example

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When you have completed the above lesson, click the next button in Canvas to progress.