2.5 Exponential function context and data modeling

Find the 10-year growth factor for each decade and compare it to the growth factor of the model.

The growth factor can be found by dividing each decade's population by the previous population.

For example, from 1900-1910: \dfrac{92.2}{76.2}=1.210

The growth factor of the model is 1.136 which is larger than the growth rates from 1930-1940 and 1960-1990, but less than the growth rates from 1900-1930, 1940-1950, and 1990-2000.

The growth factor in the model appears to be an average of the growth rates across the century. The decade from 1930-1940 had a very small growth rate which is likely due to the Great Depression. What other historical events may have influenced population growth?

Determine the decades in which the model is the least accurate.

We will use the function f(x)=76.2(1.136)^\frac{t}{10} to complete a table to estimate the population. Then, we can find the difference between the estimated population and the given population.

The model underestimates the population in every decade except 1900. The least accurate decade is 2000, when the population estimate is off by 36 million.

A graph may be a better model to visualize both the data and the function.

Explain how the model could be modified to be more accurate in predicting the population after the year 2000.

We can see that the model loses accuracy for later decades as it underestimates those populations the most. The graph needs to grow faster, so we should try a higher growth rate.

Let's try a slightly higher growth factor of 1.2. The new model would be: f(x)=76.2(1.2)^\frac{t}{10}

We can graph the original data and the modeling function using technology:

We can see this new model overestimates the data, so our growth factor is too high. Using technology, we can change the growth factor until we get a close fit for the data. A better function model would be f(x)=76.2(1.15)^\frac{t}{10}

We can see this in the graph:

There are other possible models that may model the data. We could have chosen to use only the data after the baby boom in 1950, changing both the initial value and the growth factor. Or, we could have used statistical software that uses matheatical calculations to find the best fit curve.

Summarize the problem Aziz is investigating.

A summary can include some background information, a clear problem statement, and any variable definitions that may be used to model the problem.

According to endhomelessness.org, in 2020, 580\,466 people out of the 329.5 million living in the United States experienced homelessness. However, not all of these individuals and families were unsheltered. We want to investigate how many people may be experiencing homelessness in the next ten years and how many beds will be available to help them.

In order to determine if the number of beds available will match the number of beds needed and to track people experiencing homelessness in general, we will use the following variable definitions:

• Let x represent the number of years since 2020.
• Let H(x) represent the number of people experiencing homelessness as a function of x.
• Let B(x) represent the number of available beds as a function of x.

There are many ways to plan this summary; this is just one example.

Aziz developed the following models as a part of his solution:

• H(x)=580466(1.027)^x
• B(x)=941871(1.033)^x

Write an analysis and conclusion that Aziz could use in his report.

In our analysis, we should predict how the number of people experiencing homelessness and the number of beds will change over time. In our conclusion, we need to decide what this means for the future and what our next steps should be.

Through research, we've determined that while homelessness over the last decade has been declining, in recent years, the rate of people experiencing homelessness has actually increased. We are focusing on data from 2018-2020 only for this analysis.

Predicting the future using only 3 years of data is risky, as the rates are likely to change. We will assume that the growth rates stay constant but also consider what would happen if they didn't in a worst-case scenario.

The number of beds in 2020 exceeds the homeless population and the data shows that the number of beds is increasing at a faster rate (3.3\%) than the number of people experiencing homelessness (2.7\%). If nothing changes, we will continue to have enough beds.

However, if the number of people experiencing homelessness started to grow at a faster rate as the headline implied, it would have to triple in rate to 8.1\% for the number of beds to run out, as seen in the following graph:

In conclusion, the number of beds across the U.S. should be enough to support the number of people experiencing homelessness.

There are a few factors we ignored in the simplicity of our model and how we summarized the problem for Aziz. Although across the U.S. there are enough beds, that doesn't mean that those beds are available where the people need them. Looking at a map of the population of people experiencing homelessness, we see that most people are on the west coast, in Hawaii, or in New York:

If the beds are spread out across the United States, there is a good chance that some states don't have enough beds and others have too many. Aziz should reanalyze his model and consider rates by state instead of the country as a whole.