Marko currently has [latex]20[/latex] tulips in his yard. Each year he plants [latex]5[/latex] 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 [latex]3[/latex] new albums to keep her collection current. She currently owns [latex]450[/latex] 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 [latex]P_n = P_{n-1} + 15[/latex] with initial population [latex]P_0 = 40[/latex].
Calculate [latex]P_1[/latex] and [latex]P_2[/latex].
Find an explicit formula for [latex]P_n[/latex].
Use your formula to predict the store’s sales in [latex]10[/latex] years
When will the store’s sales exceed [latex]$100,000[/latex]?
The number of houses in a town has been growing according to the recursive rule [latex]P_n = P_{n-1} + 30[/latex], with initial population [latex]P_0 = 200[/latex].
Calculate [latex]P_1[/latex] and [latex]P_2[/latex].
Find an explicit formula for [latex]P_n[/latex].
Use your formula to predict the number of houses in [latex]10[/latex] years
When will the number of houses reach [latex]400[/latex] houses?
A population of beetles is growing according to a linear growth model. The initial population (week 0) was [latex]P_0=3[/latex], and the population after [latex]8[/latex] weeks is [latex]P_8=67[/latex].
Find an explicit formula for the beetle population in week [latex]n[/latex].
After how many weeks will the beetle population reach [latex]187[/latex]?
The number of streetlights in a town is growing linearly. Four months ago [latex]n = 0[/latex], there were [latex]130[/latex] lights. Now [latex]n=4[/latex], there are [latex]146[/latex] lights. If this trend continues,
Find an explicit formula for the number of lights in month [latex]n[/latex]
How many months will it take to reach [latex]200[/latex] lights?
Tacoma’s population in 2000 was about [latex]200[/latex] thousand, and had been growing by about [latex]9\%[/latex] 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 [latex]400[/latex] thousand?
Portland’s population in 2007 was about [latex]568[/latex] thousand, and had been growing by about [latex]1.1\%[/latex] 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 [latex]700[/latex] thousand?
Diseases tend to spread according to the exponential growth model. In the early days of AIDS, the growth rate was around [latex]190\%[/latex]. In 1983, about [latex]1700[/latex] 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 [latex]5[/latex] billion and the annual growth rate was estimated at [latex]2[/latex] 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 [latex]300[/latex] bacteria. After [latex]4[/latex] hours, the population has grown to [latex]500[/latex] 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 [latex]1[/latex] 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 [latex]200[/latex] wolves were transplanted. After [latex]3[/latex] years, the population had grown to [latex]270[/latex] 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 [latex]10[/latex] years?
If this trend continues, how long will it take the population to grow to [latex]1000[/latex] wolves?
In 1968, the U.S. minimum wage was [latex]$1.60[/latex] per hour. In 1976, the minimum wage was [latex]$2.30[/latex] per hour. Assume the minimum wage grows according to an exponential model where [latex]n[/latex] 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 [latex]$5.15[/latex] in 1996, is this above, below or equal to what the model predicts?
The population of a small town can be described by the equation [latex]P_n = 4000+70n[/latex], where [latex]n[/latex] 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 [latex]P_n = 4000(1.04)n[/latex], where [latex]n[/latex] is the number of years after 2005. Explain in words what this equation tells us about how the population is changing.
A new truck costs [latex]$32,000[/latex]. 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 [latex]$24,500[/latex] after three years, write an explicit formula for the value of the car after [latex]n[/latex] 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 [latex]5\%[/latex] each year. In other words, if you have [latex]$1[/latex] this year, next year it will only buy you [latex]$0.95[/latex] worth of stuff. How much will [latex]$100[/latex] buy you in [latex]20[/latex] years?
Suppose that you have a bowl of [latex]500[/latex] M&M candies, and each day you eat [latex]\frac{1}{4}[/latex] 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 [latex]n[/latex] days.
A warm object in a cooler room will decrease in temperature exponentially, approaching the room temperature according to the formula where [latex]T_n[/latex] is the temperature after [latex]n[/latex] minutes, [latex]r[/latex] is the rate at which temperature is changing, [latex]a[/latex] is a constant, and [latex]Tr[/latex] 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 [latex](n=0)[/latex] it was [latex]85[/latex] degrees. After [latex]20[/latex] minutes, the temperature was measured again to be [latex]80[/latex] degrees. The body was in a [latex]70[/latex] 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 [latex]98.6[/latex] 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 [latex]10\%[/latex] each year, but every year [latex]100[/latex] fish are harvested from the lake by people fishing.
Write a recursive equation for the number of fish in the lake after [latex]n[/latex] years.
Calculate the population after [latex]1[/latex] and [latex]2[/latex] 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?
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?
