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8. Mathematical Modeling
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United States of America
Grades 9-12

8.04 Modeling with functions

Lesson
Practice

Introduction

We modeled with linear, exponential, and quadratic functions in Algebra 1. In this chapter, we compared different representations of functions that model real-world contexts and their key features, then used familiar and new tools to combine functions, and solved non-linear systems of equations with the various functions we learned about throughout Algebra 2. We continue to model real-world contexts in this lesson with the modeling cycle:

A modeling cycle. Starting with the phrase Identify the problem inside a circle. An arrow pointing to the right where the phrase Create a model is inside a rectangle. Next is an arrow pointing downward where the phrase Apply and analyze is inside a rectangle. Then, an arrow pointing to the right where the phrase Interpret results is inside a rectangle. Next is an arrow pointing upward where the phrase Verify the model is inside a triangle. Then, an arrow pointing to the right where the phrase Report findings is inside a circle. There is an arrow pointing to the left from the phrase Verify the model to Create a model.

Each time we model a real-world situation, we should:

  1. Identify the essential features of the problem

  2. Create a model using a diagram, graph, table, equation or expression, or statistical representation

  3. Analyze and use the model to find solutions

  4. Interpret the results in the context of the problem

  5. Verify that the model works as intended and improve the model as needed

  6. Report on our findings and the reasoning behind them

Modeling with functions

Throughout Algebra 1, Geometry, and Algebra 2, we have discussed each of the factors that make up the modeling cycle and have applied them to solve real-world problems.

Recall that we begin the modeling cycle by clarifying the problem. To do this, we:

  • Restate the problem to clarify what the model intends to measure, predict, and/or solve

  • Identify questions that need to be considered

  • List factors that will affect the outcome

  • Research information needed to answer the problem

  • State assumptions to narrow the focus

Next, the model is created. Models can be presented as equations, graphs, tables, and diagrams. Certain types of functions relate to certain types of real-world situations.

We have studied the following function types: Linear (including constant and absolute value), exponential, polynomial (including quadratic and cubic), rational, trigonometric, radical, logarithmic.

After creating a model, it is important to analyze whether the model accurately represents the situation in context.

If the model is not an accurate representation, then we need to adjust the model or decide on a new type of model.

Then, we interpret the results that come from applying the model. Interpreting results includes the following:

  • Determining whether the answer makes sense in terms of the context

  • Identifying any extraneous solutions from the model that do not apply to the situation

Next, we must verify that the model works as intended, discussing its potential accuracy depending on the limitations of the model or assumptions we made when creating the model.

If the model does not work as intended or is not accurate, we adjust and improve it as needed. If the model works and is accurate, we draw any relevant conclusions to the context.

When we make conclusions based on results, we should be prepared to report our findings. Reporting with a model includes:

  • Information that is relevant to the audience

  • Enough detail that the audience understands the reasoning behind any recommendations

Reporting a model does not include:

  • Technical algebraic work

  • Mathematical jargon that could confuse the audience

Examples

Example 1

Rainfall analysis is important to help manage and prevent flooding. The amount of rainfall during a storm in Pierson, FL was recorded over a two-hour period, and the data is displayed below:

Time in minutesRain in inches
150.067
300.115
450.213
600.347
750.933
901.062
1051.129
1201.158

Suppose we need to find the time interval when most rain accumulated and how much accumulated during that time.

a

Identify the problem. State any assumptions.

Worked Solution
Create a strategy

To identify the problem, we need to clarify what the model intends to measure, predict, and/or solve. Some questions that we need to consider are:

  • When did the storm begin and end, and how do those times relate to the times when the data was collected?

  • How will "most rain accumulated" be defined?

  • Is the amount of rain that accumulated a cause for concern?

Since we cannot research this specific storm, assumptions need to be made about when the storm began and ended and how these times relate to the times when the data collected. We also need to make an assumption about what "most rain accumulated" means.

Apply the idea

Let's assume that the data was collected shortly after the storm began, and that the storm ended shortly after two hours. More specifically, we will assume the storm begins at time zero, so there were zero inches of rain when the storm started.

The phrase "the time interval when most rain accumulated" assumes there was a time in the storm when the rain accumulated faster than other times during the storm. As we model and analyze the data, we will need to determine when it started raining harder and when the rain lightened up.

After deciding the specific times when we think it started raining harder and when the rain lessened, we will calculate the specific amount of rain that accumulated during this time interval.

Reflect and check

Remember that the assumptions we made above and the ones we will make during the modeling process will affect the final result. Others may make different assumptions that will lead to different results.

b

Create a new model to represent the data.

Worked Solution
Create a strategy

We were given data in a table, but this format may not help us determine specific times when the rain accumulated faster or slower. Instead, we will begin by plotting the data on a coordinate plane, then use the graph to help us create an equation model.

Apply the idea

Plotting the data points results in the following graph:

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x
0.1
0.2
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1
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R

Looking at the shape of the data, it appears that a cube root function would be a good model since the rain accumulates slowly, then accumulates quickly, then slows down again. We can transform the parent function y=\sqrt[3]{x} until we find an equation that appears to fit the data well.

The amount of rainfall from this particular storm can be modeled by the equation R=0.15\sqrt[3]{x-65}+0.6 where x is time in minutes and R is total rainfall in inches.

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c

Apply the model to find the time interval when most rain accumulated and how much accumulated during that time. Interpret the results.

Worked Solution
Create a strategy

At this point, we will need to decide when we think the rain started accumulating more than it did at the beginning of the storm and when it starts to slow down again. It is important that we highlight this assumption when stating the results.

Since cube root functions are symmetric about the inflection point in the center of the graph, we should choose an interval such that the middle value of the interval is the center point of the graph.

Apply the idea

According to our model, the center of the data is at x=65. It appears that most of the rain started accumulating about 10 minutes before and 10 minutes after this point, which was between 55 minutes and 75 minutes into the storm.

Substituting these values into the equation model from part (b) gives us:

\displaystyle R\displaystyle =\displaystyle 0.15\sqrt[3]{55-65}+0.6
\displaystyle \approx\displaystyle 0.2768
\displaystyle R\displaystyle =\displaystyle 0.15\sqrt[3]{75-65}+0.6
\displaystyle \approx\displaystyle 0.9232

Next, we will subtract the values to determine how much rain accumulated over that time period: 0.9232-0.2768=0.6464

Therefore, about 0.65 inches of rain accumulated between 55 minutes and 75 minutes into the storm.

This tells us that, although the storm lasted around 2 hours in total, it rained the hardest during a 20 minute period in the middle of the storm.

Reflect and check

These results are based on an assumption about when the rain accumulated faster and when it slowed down. Others may have assumed a different time interval, larger or smaller, which would have produced different results.

d

Verify the model works as intended and improve as needed.

Worked Solution
Create a strategy

To verify the model, we need to determine whether the model answers the question we need it to answer and whether it answers the question accurately. We will also need to discuss the limitations of the model.

Apply the idea

The model was designed to determine the interval when most rain accumulated, and it does satisfy this need. However, the model is limited to the data collected on this particular storm. This model cannot be used to analyze rainfall from any storm in general.

This model is also limited to the data collection process used, and data was collected during 15-minute intervals. This means there is a lack of specific data on how rain accumulated during the most intense part of the storm.

Although the shape of the model may not be completely accurate between 60 and 75 minutes into the storm, the results about when most of the rain accumulated is still fairly accurate.

e

Create a report about the accumulation of rain during the storm.

Worked Solution
Create a strategy

In our report, we need to describe how the rain accumulated during the storm, specifically noting the interval during which most rain accumulated and how much rain accumulated during that period of time.

We can also research the levels of rain that lead to flooding and include this information when creating our final report.

Apply the idea

According to www.britannica.com, rain is classified in the following ways:

  • Light - less than 0.1\text{ in.} per hour
  • Moderate - between 0.1\text{ in.} and 0.3\text{ in.} per hour
  • Heavy - more than 0.3\text{ in.} per hour

So, for the first 55 minutes of the storm, the residents of Pierson, FL experienced moderate rainfall. During the next 20 minutes of the storm, there was heavy rainfall. This 20-minute interval is when majority of the rain accumulated, amounting to about 0.65 inches. After this, there was moderate rainfall for the remaining 45 minutes of the storm.

According to www.floridadep.gov, large amounts of rain over a short period of time or large amounts of rain over a long period of time can cause flooding. However, just over 1 inch of rain that accumulated over a two-hour time span is not a relatively large amount of rain.

f

Suppose the storm did not stop after two hours, but continued to rain steadily for another hour. By the end of the third hour, a total of 1.452 inches of rainfall had accumulated.

With this new information, determine whether your model would need to be adjusted and if it does need to be adjusted, describe how.

Worked Solution
Create a strategy

We will need to determine how much rain would have accumulated according to our current model and if this is consistent with the new data. If it is not, we will need to adjust our model.

Apply the idea

Using our current model to determine the amount of rain that would accumulate after 3 hours, we get:

\displaystyle R\displaystyle =\displaystyle 0.15\sqrt[3]{180-65}+0.6
\displaystyle \approx\displaystyle 1.329

This is not consistent with the new data, so we will need to adjust the model. Since the cube root model accurately represents the first two hours of the storm, we can use a piecewise function model and determine a new equation that models the third hour of the storm.

Let's assume "steadily" refers to a linear increase in the amount of rain accumulated. We need to find the equation of the line that passes through the points \left(120, 1.158\right) and \left(180,1.452\right) representing the amount of rain accumulated between 2 and 3 hours of the storm.

\displaystyle m\displaystyle =\displaystyle \frac{y_2-y_1}{x_2-x_1}Equation for slope
\displaystyle =\displaystyle \frac{1.452-1.158}{180-120}Substitute x- and y-values
\displaystyle =\displaystyle 0.0049Evaluate
\displaystyle y\displaystyle =\displaystyle mx+bSlope-intercept form
\displaystyle 1.452\displaystyle =\displaystyle 0.0048\left(180\right)+bSubstitute slope and x- and y-values
\displaystyle 0.57\displaystyle =\displaystyle bEvaluate and subtract
\displaystyle y\displaystyle =\displaystyle 0.0049x+0.57Equation of the line

Therefore, the piecewise function that models the amount of rain accumulated over the 3 hour storm is R=\begin{cases}0.15\sqrt[3]{x-65}+0.6, & 0\leq x<120 \\0.0049x+0.57, & x\geq 180\end{cases}

Reflect and check

To check how well our piecewise function models the known data, we can graph it with the known data values.

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R

The equations appear to model the data well. As more data is collected, models will often need to be adjusted so they more accurately represent the context.

Idea summary

When modeling real-world contexts, we want to remember that models are not perfect, but they provide an opportunity for us to interpret the world around us. Each time we model a real-world situation, we should:

  1. Identify the essential features of the problem.

  2. Create a model using a diagram, graph, table, equation or expression, or statistical representation.

  3. Analyze and use the model to find solutions.

  4. Interpret the results in the context of the problem.

  5. Verify that the model works as intended and improve the model as needed.

  6. Report on our findings and the reasoning behind them.

Outcomes

A.CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.C.9

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.BF.A.1

Write a function that describes a relationship between two quantities.

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