# 8.7: Exercises

- Page ID
- 34224

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## Skills

- Marko currently has 20 tulips in his yard. Each year he plants 5 more.
- Write a recursive formula for the number of tulips Marko has
- Write an explicit formula for the number of tulips Marko has

- Pam is a Disc Jockey. Every week she buys 3 new albums to keep her collection current. She currently owns 450 albums.
- Write a recursive formula for the number of albums Pam has
- Write an explicit formula for the number of albums Pam has

- A store’s sales (in thousands of dollars) grow according to the recursive rule \(P_n=P_{n-1} + 15\), with initial population \(P_0=40\).
- Calculate \(P_1\) and \(P_2\)
- Find an explicit formula for \(P_n\)
- Use your formula to predict the store’s sales in 10 years
- When will the store’s sales exceed $100,000?

- The number of houses in a town has been growing according to the recursive rule \(P_n=P_{n-1} + 30\), with initial population \(P_0=200\).
- Calculate \(P_1\) and \(P_2\)
- Find an explicit formula for \(P_n\)
- Use your formula to predict the number of houses in 10 years
- When will the number of houses reach 400 houses?

- A population of beetles is growing according to a linear growth model. The initial population (week 0) was \(P_0=3\), and the population after 8 weeks is \(P_8=67\).
- Find an explicit formula for the beetle population in week \(n\)
- After how many weeks will the beetle population reach 187?

- The number of streetlights in a town is growing linearly. Four months ago \((n = 0)\) there were 130 lights. Now \((n = 4)\) there are 146 lights. If this trend continues,
- Find an explicit formula for the number of lights in month \(n\)
- How many months will it take to reach 200 lights?

- Tacoma's population in 2000 was about 200 thousand, and had been growing by about 9% each year.
- Write a recursive formula for the population of Tacoma
- Write an explicit formula for the population of Tacoma
- If this trend continues, what will Tacoma's population be in 2016?
- When does this model predict Tacoma’s population to exceed 400 thousand?

- Portland's population in 2007 was about 568 thousand, and had been growing by about 1.1% each year.
- Write a recursive formula for the population of Portland
- Write an explicit formula for the population of Portland
- If this trend continues, what will Portland's population be in 2016?
- If this trend continues, when will Portland’s population reach 700 thousand?

- Diseases tend to spread according to the exponential growth model. In the early days of AIDS, the growth rate was around 190%. In 1983, about 1700 people in the U.S. died of AIDS. If the trend had continued unchecked, how many people would have died from AIDS in 2005?

- The population of the world in 1987 was 5 billion and the annual growth rate was estimated at 2 percent per year. Assuming that the world population follows an exponential growth model, find the projected world population in 2015.

- A bacteria culture is started with 300 bacteria. After 4 hours, the population has grown to 500 bacteria. If the population grows exponentially,
- Write a recursive formula for the number of bacteria
- Write an explicit formula for the number of bacteria
- If this trend continues, how many bacteria will there be in 1 day?
- How long does it take for the culture to triple in size?

- A native wolf species has been reintroduced into a national forest. Originally 200 wolves were transplanted. After 3 years, the population had grown to 270 wolves. If the population grows exponentially,
- Write a recursive formula for the number of wolves
- Write an explicit formula for the number of wolves
- If this trend continues, how many wolves will there be in 10 years?
- If this trend continues, how long will it take the population to grow to 1000 wolves?

- One hundred trout are seeded into a lake. Absent constraint, their population will grow by 70% a year. The lake can sustain a maximum of 2000 trout. Using the logistic growth model,
- Write a recursive formula for the number of trout
- Calculate the number of trout after 1 year and after 2 years.

- Ten blackberry plants started growing in my yard. Absent constraint, blackberries will spread by 200% a month. My yard can only sustain about 50 plants. Using the logistic growth model,
- Write a recursive formula for the number of blackberry plants in my yard
- Calculate the number of plants after 1, 2, and 3 months

- In 1968, the U.S. minimum wage was $1.60 per hour. In 1976, the minimum wage was $2.30 per hour. Assume the minimum wage grows according to an exponential model where n represents the time in years after 1960.
- Find an explicit formula for the minimum wage.
- What does the model predict for the minimum wage in 1960?
- If the minimum wage was $5.15 in 1996, is this above, below or equal to what the model predicts?

## Concepts

- The population of a small town can be described by the equation \(P_n = 4000 + 70n\), where \(n\) is the number of years after 2005. Explain in words what this equation tells us about how the population is changing.

- The population of a small town can be described by the equation \(P_n = 4000(1.04)n\), where \(n\) is the number of years after 2005. Explain in words what this equation tells us about how the population is changing.

## Exploration

Most of the examples in the text examined growing quantities, but linear and exponential equations can also describe decreasing quantities, as the next few problems will explore.

- A new truck costs $32,000. The car’s value will depreciate over time, which means it will lose value. For tax purposes, depreciation is usually calculated linearly. If the truck is worth $24,500 after three years, write an explicit formula for the value of the car after \(n\) years.

- Inflation causes things to cost more, and for our money to buy less (hence your grandparents saying, "In my day, you could buy a cup of coffee for a nickel"). Suppose inflation decreases the value of money by 5% each year. In other words, if you have $1 this year, next year it will only buy you $0.95 worth of stuff. How much will $100 buy you in 20 years?

- Suppose that you have a bowl of 500 M&M candies, and each day you eat \(\frac{1}{4}\) of the candies you have. Is the number of candies left changing linearly or exponentially? Write an equation to model the number of candies left after n days.

- A warm object in a cooler room will decrease in temperature exponentially, approaching the room temperature according to the formula where Tn is the temperature after n minutes, \(r\) is the rate at which temperature is changing, a is a constant, and Tr is the temperature of the room. Forensic investigators can use this to predict the time of death of a homicide victim. Suppose that when a body was discovered \((n = 0)\) it was 85 degrees. After 20 minutes, the temperature was measured again to be 80 degrees. The body was in a 70 degree room.
- Use the given information with the formula provided to find a formula for the temperature of the body.
- When did the victim die, if the body started at 98.6 degrees?

- Recursive equations can be very handy for modeling complicated situations for which explicit equations would be hard to interpret. As an example, consider a lake in which 2000 fish currently reside. The fish population grows by 10% each year, but every year 100 fish are harvested from the lake by people fishing.
- Write a recursive equation for the number of fish in the lake after n years.
- Calculate the population after 1 and 2 years. Does the population appear to be increasing or decreasing?
- What is the maximum number of fish that could be harvested each year without causing the fish population to decrease in the long run?

- The number of Starbucks stores grew after first opened. The number of stores from 1990-2007, as reported on their corporate website[1], is shown below.
- Carefully plot the data. Does is appear to be changing linearly or exponentially?
- Try finding an equation to model the data by picking two points to work from. How well does the equation model the data?
- Try using an equation of the form , where k is a constant, to model the data. This type of model is called a Power model. Compare your results to the results from part \(b\).
*Note:**to use this model, you will need to have 1990 correspond with*\(n = 1\)*rather than*\(n = 0\).

- Thomas Malthus was an economist who put forth the principle that population grows based on an exponential growth model, while food and resources grow based on a linear growth model. Based on this, Malthus predicted that eventually demand for food and resources would out outgrow supply, with doom-and-gloom consequences. Do some research about Malthus to answer these questions.
- What societal changes did Malthus propose to avoid the doom-and-gloom outcome he was predicting?
- Why do you think his predictions did not occur?
- What are the similarities and differences between Malthus's theory and the logistic growth model?

[1] www.starbucks.com/aboutus/Company_Timeline.pdf retrieved May 2009