a. Compare the average rate of change from 1950 to 1970 for both the U.S. and world populations.

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Jeffry Carroll
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1 Aim #84: How do we compare linear and exponential growth? Homework: Handout Do Now: Callie and Joe are examining the population data in the graphs below for a history report. Their comments are as follows: Callie: It looks like the U.S. population grew the same amount as the world population, but that can t be right, can it? Joe: Well, I don t think they grew by the same amount, but they sure grew at about the same rate. Look at the slopes. a. Compare the average rate of change from 1950 to 1970 for both the U.S. and world populations. b. Do the graphs above seem to indicate linear or exponential population growth? c. Write a linear equation that models the world population growth from based on the information in the graph. Assume that the population (in millions) in 1950 was 2,500 and in 2000 was 6,000. Use x to represent the number of years after 1950.

4 (Continued from last page) d. How is this graph similar to the World Population graph in Example 1? How is it different? e. Does the behavior of the graph from match that shown on our last graph? f. Why is the first graph somewhat misleading? 2) Two social media websites open their memberships to the public. a. Compare the websites' memberships by calculating and interpreting the average rate of change from Day 10 to Day 20. b. Does the graph and the chart indicate a linear or exponential growth? c. Which website will have more members after 50 days? Explain.

5 3) In 1900, Littleton had a population of 1000 people. Littleton's population increased by 50 people each year. In 1900, Tinyville had a population of 500 people. Tinyville's population increased by 5% each year. a. Let x represent the number of years since Write a function to model the population in each town. b. During what year does Tinyville's population exceed Littleton's population? Justify your answer. c. Suppose Littleton's initial population doubled to 2000 and maintained a constant rate of increase of 50 people each year. Did Tinyville's population still catch up to Littleton's population? If so, during what year? 4) The table below represents the population of the U.S. (in millions) for the specified years. a. If we use the data from to create an exponential equation representing the population, we generate the following formula for the sequence, where f(t) represents the U.S. population and t represents the number of years after f(t) = 5(1.0204) t. Use this formula to determine the population of the U.S. in the year b. If we use the data from to create an exponential formula that models the population, we generate the following, where f(t) represents the U.S. population and represents the number of years after f(t) = 76(1.013) t. Use this formula to determine the population of the U.S. in the year 2010.

6 5) After graduation, you are offered two jobs. Cedar Grove Associates offered to start you at $30,000 with a 6% increase per year. Maple Grove Associates offered to start you at $40,000 with a $1200 raise per year. a. Write a formula to model the salary offer of Cedar Grove Associates. Is your formula linear or exponential? b. Write a formula to model the salary offer of Maple Grove Associates. Is your formula linear or exponential? c. If you plan on keeping the job for only 5 years, which company would be the better option for you to choose? Explain. d. If you intend to remain with the company, which company would be the better option to choose as a long term career? Explain.

7 6) The population of the country of Oz was 600,000 in the year The population is expected to grow by a factor of 5% annually. The annual food supply of Oz is currently sufficient for a population of 700,000 people and is increasing at a rate which will supply food for an additional 10,000 people per year. a. Write a formula to model the population of Oz. Is your formula linear or exponential? b. Write a formula to model the food supply. Is the formula linear or exponential? c. At what point does the population exceed the food supply? Justify your response. d. If Oz doubled its current food supply (to 1.4 million), would shortages still take place? Explain. e. If Oz doubles both its beginning food supply and doubles the rate at which the food supply increases, would food shortages still take place? Explain. Sum it up! -Graphs can be misleading. Depending on the time interval, data that appears linear on one time interval may be non-linear when a larger time span is considered. -Linear: constant rates over equal intervals. -Exponential: constant factors over equal intervals.

Lesson 6: Exponential Growth U.S. Population and World Population

Lesson 6: Exponential Growth U.S. Population and World Population Exponential Growth U.S. Population and World Population Classwork Mathematical Modeling Exercise 1 Callie and Joe are examining the population data in the graphs below for a history report. Their comments