Population Growth

Group Problem Begins Here: Population Growth 

As a city's population grows, so does its need for basic infrastructure and resources, such as roads and schools, electricity and water, etc. That's why most cities, try to manage their growth through careful planning. Property developers and home builders on the other hand often have no direct ties to the community and have little incentive to restrain growth. Thus, the city management team and property developers/builders frequently find themselves at odds.

According to the most recent census bureau (2020), Deerfield Beach (Links to an external site.) - a city in Broward County - currently has a population of 81,915 people. With this information, the city council must develop a ten year growth plan that maintains adequate resources and infrastructure for the expanding population. Planners on the city commission have said that a conservative growth rate of 4% per year is preferable. Property developers/builders however have been lobbying for a much more liberal growth rate of 8.5% per year, claiming that the increase in population will bring in additional taxes that will help offset the cost of the additional resources and infrastructure needed to support the expanding population. 

Write two functions to represent the two growth plans. Let C(t)C(t) be the population after tt years with the commission's proposed growth rate and let D(t)D(t) be the population under the property developers/builders plan. With each function, determine what the population would be after 1 year, 2 years, .... 10 years. Then, with each function, determine in what year the population would double. 

Before making a final decision on the growth plan for Deerfield Beach, the city council decided to hire an independent research team. The research team concluded that with the city's current tax rate, the city can raise enough revenue to maintain it's infrastructure and resources for a total population of 120,000 people. However, if the population exceeded 120,000 people, then the taxes would need to be raised; which the city would like to avoid (higher taxes usually causes a population decline, as city residents/businesses will move out of the city to avoid high taxes). With the goal of not exceeding 120,000 people, determine which growth plan (or both) would be acceptable. 

Your solution MUST include responses to ALL four parts.

1. Understand the problem.

· Restate the problem in your own words.

· What do you know from the reading the problem? *Try to use the mathematical language we've been learning about this week.  

· What type of mathematical function is this? Can you create the function from what you know already?

· What is needed in order to move forward with answering the two questions above? *Try making a KWL chart, a mind-map or a flow chart. 

2. Make a plan.

State your plan for finding a possible answer to the questions above. You may use words, diagrams or algebraic calculations. You might want to consider making a table, looking for a pattern, or building an equation or graphical model.

3. Implement your plan.

Once you have articulated your plan, carry out your plan.

Using the information given, create a mathematical model (an equation) for each proposed growth rate that you can use to determine the answers to the questions above. In your mathematical model, make sure to specify what your variables represent, determine how you are measuring your variables (hours? minutes? dollars?, etc.), create a graph of your mathematical model, show your algebraic computations, etc. 

4. Look back. Is your answer reasonable? Can you find a way to check your work? Interpret your results.

Remember that you may have multiple representations – words, tables, graphs, and/or equations. Can you find another way to look at this problem that would allow you to check that your solution is correct? Interpret the answers in the context of the original application. Use complete sentences.