3.2E: Modeling with Exponential Functions (Exercises)
Section 3.2 Exercises
Part 1: Modeling with Exponential functions with a constant percentage rate of change or growth factor.
1. On Monday 20 computers are infected with a computer virus. Each day, the number of infected computers triples.
a. Find an exponential model for the number of infected computers, y, in terms of t, the days since Monday.
b. Predict the number of infected computers in 2 weeks.
2. What is the growth factor b in each of the following situations of exponential growth or decay?
a. A customer's power usage increased by 9% per year
b. The number of subscribers' decreased by 3% per year.
c. The population of a species of from increased by 2.47% per month?
3. For each of the following situations the percentage rate of change per second and initial value at time t = 0 sec. is given. Find an formula for the quantity y as a function of t. Then predict the amount in 12 seconds.
a. The initial amount is 520 ft and is increasing by 4% per second.
b. The initial amount 400 mg and is decreasing by 3% per second.
c. The initial amount is 20 m 3 and is decreasing by 2.41% per second.
4. The populations, P, of six towns, t years after 2010 are given below.
i. \(P=12,000(1.07)^{t} \) ii. \(P=8,000(1.13)^{t} \)
iii. \(P=9,000(0.94)^{t} \) iv. \(P=8,500(1.085)^{t} \)
v. \(P=8,214(0.83)^{t} \) vi. \(P=8,912(0.997)^{t} \)
a. Which towns are increasing in size? Which towns are decreasing in size?
b. Which town is growing the fastest? What is the annual percentage growth rate of that town?
c. Which town is decreasing in size the fastest? What is the annual percentage decay rate of that town?
d. Which town had the largest population in 2010?
5. As of January 2, 2022, the rate of people 70 years or older hospitalized for COVID was 13.32 people per 100,000 people in the US population. At that point in time, the number of people hospitalized was increasing by 3.7% per day . 1 Assume the rate of infection continues to increase at this percentage rate.
a. Find an exponential model for the number of people 70 years or older hospitalized , \(H(t)\) , in terms of t, the days since January 2, 2022.
b. Predict the number of people infected on Jan. 22.
6. As of February 4, 2022, the rate of people 70 years or older hospitalized for COVID was 15.52 people per 100,000 people in the US population. At that point in time, the number of people hospitalized was decreasing by 4.475% per day. 2 Assume the rate of infection continues to decrease at this percentage rate.
a. Find an exponential model for the number of people 70 years or older hospitalized , \(H(t)\) , in terms of t, the days since February 4, 2022.
b. Predict the number of people infected on February 23, 2022.
7. Clovis Community College had 10,464 students in the Fall 2016 - Spring 2017 academic year. At the time, growth was projected to be 8% per years based on state funding projections. 3 Assume the student enrollment continues to increase at this percentage rate.
a. Find an exponential model for the number of students, \(S(t)\) , in terms of the years since the Fall 2016 - Spring 2017 .
b. Predict the number of students at Clovis Community College in the Fall 2026 - Spring 2027 semester.
8. The concentration of fine particulate matter pollution (PM2.5) in California was 13.95 \(\dfrac{\mu g}{m^{3}} \) in 2003. From 2003 to 2011, the concentration decreased by approximately 4.942% per year. 4 Assume the concentration of fine particulate matter pollution continues to decrease at this percentage rate.
a. Find an exponential model for the concentration of fine particulate matter pollution, \(c(t)\) , in terms of the years since 2003.
b. Predict when the concentration of fine particulate matter pollution in 2025.
9. The fox population in a certain region has an annual growth rate of 9% per year. It is estimated that the population in the year 2010 was 23,900. Estimate the fox population in the year 2018.
10. The amount of area covered by blackberry bushes in a park has been growing by 12% each year. It is estimated that the area covered in 2009 was 4,500 square feet. Estimate the area that will be covered in 2020.
11. A vehicle purchased for $32,500 depreciates at a constant rate of 5% each year. Determine the approximate value of the vehicle 12 years after purchase.
12. A business purchases $125,000 of office furniture which depreciates at a constant rate of 12% each year. Find the residual value of the furniture 6 years after purchase.
Part 2: Modeling with exponential functions from data.
In problems 13- 17, find a formula for an exponential function passing through the two points.
13. \(\left(0,\; 6\right),\; (3,\; 750)\)
14. \(\left(0,\; 3\right),\; (2,\; 75)\)
15. \(\left(0,\; 2000\right),\; (2,\; 20)\)
16. \(\left(-1,\dfrac{3}{2} \right),\; \left(3,\; 24\right)\)
17. \(\left(3,\; 1\right),\; (5,\; 4)\)18. A house that was valued at $300,000 in California in January 2018 was valued at $376,000 in January 2021. 5
a. Find an exponential formula for the value of the house, \(v(t)\) t years since January 2018, assuming the value continues at the same percentage rate.
b. Predict the value of the home in January 2025.
c. What was the annual growth rate?
19. A house that was valued at $400,000 in California in January 2019 was valued at $480,000 in January 2022. 6
a. Find an exponential formula for the value of the house, \(v(t)\), t years since January 2019, assuming the value continues to increase at the same percentage rate.
b. Predict the value of the home in January 2025.
c. What was the annual growth rate?
20. A car was valued at $38,000 in the year 2012. The value depreciated to $11,000 by the year 2021. Assume that the car value continues to drop by the same percentage rate.
a. Find an exponential formula for the value of the car, \(v(t)\), t years since 2012.
b. Predict the value of the car in 2025.
21. A car was valued at $24,000 in the year 2006. The value depreciated to $15,000 by the year 2009. Assume that the car value continues to drop by the same percentage rate. What was the value in the year 2014?
22. The population of California was 37,253,956 on April 1, 2010. The population grew to 39,538,233 on April 1, 2020. Assume that the population continues to increase by the same percentage rate. 7
a. Find an exponential formula for the population of California, \(P(t)\), t years since April 1, 2010.
b. Predict the population of California on April 1, 2025.
23. A person drinks alcohol at a party. After her last drink, the alcohol level of her blood soon reaches a maximum of 0.28 milligram per milliliter of blood. 50% of the alcohol remained after 2 hours. Assume the blood level decreases exponentially.
a. Find an exponential formula for the alcohol level in her blood, \(A(t)\), in t hours.
b. If she waits 3 hours, will her blood alcohol level be below the legal limit of 0.08?
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1,2 The data was reported by the CDC at https://covid.cdc.gov/covid-data-tra...tal-admissions on January 20, 2023.
3 The data was r eported by the Clovis Community College Institutional Research Department at Clovis Community College on August 12, 2018 ( https://www.cloviscollege.edu/about/...-research.html ) .
4 The data was reported by the CDC at https://wonder.cdc.gov on February 24, 2023.
5,6 The data was reported by the U.S. Federal Housing Finance Agency House Price Index Calculator at www.fhfa.gov/DataTools/Tools/Pages/HPI-Calculator.aspx on January 25, 2023.
7 The data was reported by the U.S. Census Bureau at https://www.census.gov/quickfacts/fa...e/CA/PST045222 on January 26, 2023.
- Answer
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1. a. \(y=20(3)^{t} \) b. y = 95,659,380
3. a. \(y=520(1.04)^{t} \) b. \(y=400(0.97)^{t} \) c. \(y=20(0.9759)^{t} \)
5. a. \(H(t) =13.32(1.037)^{t} \) b. 27.55
7. a. \(S(t) = 10,464(1.08)^t\) b. 22,591 students
9. The fox population can be modeled by the function \(y=23,900(1.09)^{t} \). In 2018, there will be 47622 Fox.
11. The vehicle \(V=32,500(0.95)^{t} \). In 12 years, the value will be $17561.70.
13. \(y = 6(5)^x\)
15. \(y = 2000(0.1)^x\)
17. \(y = (\dfrac{1}{8})(2)^x\)
19. a. \(y = 400,000(1.0627)^{t}\) b. 576,000 c. The home value is growing by 6.27%
21. The car will be worth $6,853.
23. a. \(A(t)=0.28(0.7071)^{t}\) b. The blood alcohol level is 0.09899 after 3 hours.