Learning objectives
Once a model has been created, we need to apply it to the context and verify that it works the way we had intended. If the results from our model do not appear to be accurate, we can revisit our assumptions and repeat the modeling cycle.
Exponential functions can be difficult to identify from a graph or table when the amount of change is small relative to the size of the numbers.
Consider the data from the U.S. Census for each decade in the 20th century:
Year | Population (in millions) |
---|---|
1900 | 76.2 |
1910 | 92.2 |
1920 | 106 |
1930 | 122.8 |
1940 | 132.2 |
1950 | 150.7 |
1960 | 179.3 |
1970 | 203.3 |
1980 | 226.5 |
1990 | 248.7 |
2000 | 308.7 |
The function P(t)=76.2(1.136)^\frac{t}{10} has been created from the table to find the population t years after 1900.
Find the 10-year growth factor for each decade and compare it to the growth factor of the model.
Determine the decades in which the model is the least accurate.
Explain how the model could be modified to be more accurate in predicting the population after the year 2000.
Exponential models can be used to predict growth and decay over a specified domain. Models can be verified by testing points whose values we know and then updated using different assumptions as needed.
After we've used a model to solve a problem, it's time to write a report that summarizes the results and process. The stakeholders are the people who would be most interested in knowing the solution to the problem.
When writing a report, we generally want to include:
Consider the following problems:
For each problem, who would be the stakeholders? Give an explanation for why each problem might need to be solved.
Aziz read a headline recently that said, "Local leaders fear affordable housing crisis could lead to unprecedented homelessness crisis of 8.1 \%." He decided to investigate the problem and model the shelter needs for the next ten years.
Summarize the problem Aziz is investigating.
Aziz developed the following models as a part of his solution:
Write an analysis and conclusion that Aziz could use in his report.
Reports should be written as professionally as possible. Present the results, supported by models both visual and algebraic. Summarize how those models were created and what those models tell us about the problem. Then, explain the limits of those models and how they might be improved for further investigation.