Exercises 4.4 Exercises
1.
Let \(P_n = P_{n-1} + 18\text{.}\) If \(P_0 = 50\text{,}\) find \(P_1\text{,}\) \(P_2\text{,}\) and \(P_3\text{.}\)
2.
The population of a city is growing according to the formula,
- Complete the table of values.
Table 4.4.1. \(n\) \(P_n\) 0 457 1 2 3 4 Graph your results.
Write an explicit equation for \(P_n\text{.}\)
Use your equation to predict the population in 50 years.
3.
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.
\(\displaystyle P_0 = 20, P_n = P_{n-1} + 5\)
\(\displaystyle P_n = 20 + 5n\)
4.
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.
5.
A store’s sales (in thousands of dollars) grow according to the recursive rule \(P_n=P_{n-1} + 15\text{,}\) with initial population \(P_0=40\text{.}\)
Calculate \(P_1\) and \(P_2\text{.}\)
Find an explicit formula for \(P_n\text{.}\)
Use your formula to predict the store’s sales in 10 years.
When will the store’s sales exceed $100,000?
\(\displaystyle P_1 = 55, P_2 = 70\)
\(\displaystyle P_n = 40 + 15n\)
190 thousand dollars
n = 4 years
6.
The number of houses in a town has been growing according to the recursive rule \(P_n=P_{n-1} + 30\text{,}\) with initial population \(P_0=200\text{.}\)
Calculate \(P_1\) and \(P_2\text{.}\)
Find an explicit formula for \(P_n\text{.}\)
Use your formula to predict the number of houses in 10 years.
When will the number of houses reach 400 houses?
7.
A population of beetles is growing according to a linear growth model. The initial population (week 0) was \(P_0=3\text{,}\) and the population after 8 weeks is \(P_8=67\text{.}\)
Find an explicit formula for the beetle population in week \(n\text{.}\)
After how many weeks will the beetle population reach 187?
\(\displaystyle P_n = 3+8n\)
23 weeks
8.
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 \(n\text{.}\)
How many months will it take to reach 200 lights?
9.
The revenue of a company (in millions) n years after 2005 is given in the graph. Write an equation that models the revenue n years after 2005. 
10.
The cost of tuition \(n\) years after 2010 is shown in the graph. Write an equation that models the tuition and use your model to predict the tuition in the year 2030. 
11.
The number of miles driven after \(n\) hours is given by the equation \(M_n = 45n\text{.}\)
What is the common difference?
Write a sentence to describe what the common difference is telling us in this context. Include units.
12.
The value of a car \(n\) years after it was purchased is given by the equation \(V_n = 9500-625n\text{.}\)
What is the common difference?
Write a sentence to describe what the common difference is telling us in this context. Include units.
13.
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?
\(\displaystyle P_0 = 200, P_n = 1.09P_{n-1}\)
\(\displaystyle P_n = 200\left(1.09\right)^n\)
\(\displaystyle P_{16}=200(1.09)^{16}=794.061(\text { thousand })=794,061\)
14.
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?
15.
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?
16.
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.
17.
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?
\(\displaystyle P_0 = 300, P_n = 1.136P_{n-1}\)
\(\displaystyle P_n = 300\left(1.136 \right)^n\)
6400 bacteria
18.
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?
19.
The population of a small town can be described by the equation \(P_n = 4000 + 70n\text{,}\) where \(n\) is the number of years after 2005. Explain in words what this equation tells us about how the population is changing.
20.
The population of a small town can be described by the equation \(Pn = 4000\left(1.04 \right)^n\text{,}\) where \(n\) is the number of years after 2005. Explain in words what this equation tells us about how the population is changing.
21.
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.
22.
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?
23.
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.
24.
The table below shows the atmospheric pressure (in millibars) at various altitudes (in kilometers).
Altitude (n)
Pressure (P)
0
1013
2
795
Use the data in the table to find a linear equation that gives the pressure \(n\) kilometers above sea-level.
Use the data in the table to find an exponential equation that gives the pressure \(n\) kilometers above sea-level.
If the pressure on top of Mt Everest (which has an altitude of 8.8 km) is 325 mb, is the linear or exponential equation a better model of atmospheric pressure? Explain.
25.
A small business buys a computer for $4,000. After 4 years, the value of the computer is expected to be $200.
If the value of the computer decreased linearly, find an explicit linear equation that gives the value of the computer \(n\) years after it was purchased.
If the value of the computer decreased exponentially, find an equation that gives the value of the computer \(n\) years after it was purchased.
Use both your linear and exponential model to predict the value of the computer 6 years after it was purchased. Which model do you think is more accurate? Explain.
26.
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?
Exercise Group.
For the following exercises, write an equation describing the relationship of the given variables.
27.
\(y\) varies directly as \(x\) and when \(x=6\text{,}\)\(y=12\text{.}\)
28.
\(y\) varies directly as the square of \(x\) and when \(x=4\text{,}\) \(y = 80\text{.}\)
29.
\(y\) varies inversely as \(x\) and when \(x=4\text{,}\) \(y=2\text{.}\)
30.
\(y\) varies inverse as the square of \(x\) and when \(x=3\text{,}\) \(y=2\text{.}\)
31.
If \(y\) varies directly as \(x\) and \(y = 6\) when \(x = 2\text{,}\) what is \(y\) when \(x = 10\text{?}\)
32.
If \(v\) varies directly with the square of \(w\) and \(v = 6\) when \(w = 3\text{,}\) what is \(v\) when \(w = 4\text{?}\)
33.
If \(a\) varies inversely with the square of \(b\) and \(a = 12\) when \(b = 2\text{,}\) what is \(a\) when \(b = 4\text{?}\)
34.
If \(m\) varies directly with \(t\) and \(m = 12\) when \(t = 5\text{,}\) what is \(m\) when \(t= 8\text{?}\)
35.
If \(y\) varies inversely with \(x\) and \(y = 8\) when \(x = 2\text{,}\) what is \(y\) when \(x = 6\text{?}\)
36.
If \(y\) varies inversely with \(x\) and \(y = 3\) when \(x = 2\text{,}\) what is \(x\) when \(y = 6\text{?}\)
37.
The fuel consumption (mpg) of a car varies inversely with its weight. A car that weighs 3800 pounds gets 33 mpg on the highway. What would be the fuel consumption of a car that weighs 5500 pounds?
38.
The length a spring stretches varies directly with the weight placed at the end of the spring. When a 5-pound weight is placed on a spring, it stretches 2 inches. How far would the spring stretch if a 9-pound weight is placed on it?
39.
The distance that an object falls varies directly as the square of the time of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?
40.
The number of hours it takes ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours when the temperature is 54 degrees. How long would it take for the same block to melt if the temperature were 45 degrees?
41.
The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.1 foot-candles at a distance of 3 meters. Find the intensity at 8 meters.
42.
The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.