3.3E: More on Compounding Interest and Continuous Exponential Models (Exercises)
Section 3.3 Exercise
Part 1: More on compounding interest
1. An investment $4,000 is made in a bank account at an annual percent rate of 7% per year. Predict the amount in the account find the amount in the bank after 9 years if the interest is compounded
a. annually b. quarterly c. monthly
2. A person borrows $20,000 from a bank that charges 12% interest per year that is compounded monthly.a. Find a formula for the amount owed after t years.
b. Predict the amount owed after 5 years if none of the balance is paid.
c. What is the effective annual yield?
3 . A person invests $100,000 into a fund that pays 8% interest per year that is compounded monthly.
a. Find a formula for the amount in the fund after t years.
b. Predict the amount in the account after 5 years if none of the balance is paid.
c. What is the effective annual yield?
4. An investment grows by 6% per year over a 10 year period. By what percent does it increase over the 10-year period?
5. Find the annual percentage yield (APY) for a savings account with annual percentage rate of 3% compounded
a. quarterly b. Monthly c. daily
6. Which of the following investments is more valuable?
Investment A: Pays 5% APR compounded quarterly
Investment B: Pays 4.9% APR compounded monthly
Investment C: Pays 4.8% APR compounded daily
7. Suppose $5000 is invested into one of five accounts. Let t represent the time in years. Match each formula (A) - (E) with the description of each account (i) - (iv). Some descriptions may match more that once investment. Indicate all investments that match a particular description.
Investment A: \(A = 5000(1.16)^{t}\) Investment B: \(A = 5000(1.12)^{t}\)
Investment C: \(A = 5000(1.08)^{2t}\) Investment D: \(A = 5000(1.01)^{12t}\)
Investment E: \(A = 5000(1.04)^{4t}\)
(i) This investment earned 12% interest per year compounded monthly.
(ii) This investment earned 16% interest per year compounded quarterly.
(iii) This investment earned more than 16% interest per year compounded annually.
(iv) This investment earned less than 4% interest per quarter.
Part 2: Continuous exponential growth and decay
8. Without a graphing utility such as a calculator or app, match the functions with a graph
(i) \(y\; =\; 5e^{0.76t}\) (ii) \(y\; =\; 5e^{-0.34t}\) (iii) \(y\; =\; 5e^{0.23t}\) (iv) \(y\; =\; 5e^{-0.85t}\)
9. For each formula below (where t represents the time in years from now), determine
(a) What is the initial value at time t = 0?
(b) Is the quantity increasing or decreasing?
(c) Is the percentage growth rate continuous or not?
(d) What is the percent growth or decay rate?
Formula 1: \(y\; =\; 20e^{1.27t}\) Formula 2: \(y\; =\; 3.4(1.27)^{t}\)
Formula 3: \(y\; =\; 1.29e^{-0.48t}\) Formula 4: \(y\; =\; 3.4(0.58)^{t}\)
10. The population of a town is currently 12,000 people. In each scenario, find a formula for the population in t years. Then predict the population in 12 years.
(a) Scenario 1: The population is growing continuously at 14% .
(b) Scenario 2: The population is growing periodically by 14% per year.
11. A petri dish contains 1500 bacteria, when an acidic solution is applied. In each scenario, find a formula for the number of bacteria in t hours. Then predict the number of bacteria in 5 hours.
(a) Scenario 1: The number of bacteria decreases periodically by 27% per hour.
(b) Scenario 2: The number of bacteria decreases continuously by 27% per hour.
12. A wildfire has burned 4,257 acres. Assume that the fire continues to expand at a continuous rate of 64% per day.
a. Find a formula for the number of acres burned \(A(t)\) in t days.
b. Predict the number of acres burned in two weeks.
c. Use a graphing utility to predict when the number of acres burned will reach 20,000 acres.
13. In an elk habitat, there are currently 12,571 acres contain elk in 2012. Due to environmental and other pressures the number of acres containing elk decreased continuously by 8.9% in the next year. 1 Assume that the acreage containing elk to expand at this percentage rate.
a. Find a formula for the acres containing elk \(A(t)\) in t years.
b. Predict the acres containing elk in 10 years.
c. Use a graphing utility to predict when the number of acres containing elk is only 1,000.
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1 - The data was reported by the US Fish and Wildlife service at https://www.fws.gov/data/resources on February 2,2023.
- Answer
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1. a. $7,353.84 b. $7,469.63 c. $7,496.713. a. \(A = 100,000(1.00667)^{12t}\) b. $148,987.53 c. 8.2996%
5. a. 3.0339% b. 3.0416% c. 3.0453%
7. (i) - D , (ii) - E, (iii) - C & E, (iv) - A & D
9. Formula 1: The initial value a = 20. The function is increasing by 27% per year. The growth is continuous.
Formula 2: The initial value a = 3.4. The function is increasing by 27% per year. The growth is not continuous.
Formula 3: The initial value a = 1.29. The function is decreasing by 48% per year. The growth is continuous.
Formula 4: The initial value a = 3.4. The function is decreasing by 48% per year. The growth is not continuous.
11. a. \(A = 1,500(0.73)^{t}\). After 5 years, there are 4,956 bacteria.
b. \(A = 1,500(e)^{0.27t}\). After 5 years, there are 5,786 bacteria.
13. a. \(A = 12,571(e)^{-0.089t}\) b. 5,162 acres c. 28.4 years