Chapter 8: Exponential Word Problems

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1 Dynamics of Algebra 2 Name: Date: Block: Chapter 8: Exponential Word Problems E XPONENTIAL G ROWTH Exponential Growth Formula: a = r = t = 1. A host is a computer that stores information you can access through the Internet. For example, Web sites are stored on host computers. In January, 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. a. Write a model giving the number h (in millions) of hosts t years after b. About how many hosts were there in 1996? Growth Factor vs. Percent Increase In the previous example, notice that the annual percent increase was 100%. This translated into a growth factor of 2, which means that the number of Internet hosts doubled each year. People often confuse percent increase and growth factor, especially when a percent increase is 100% or more. For example, a percent increase of 200% means that a quantity tripled, because the growth factor is = 3. When you hear or read reports of how a quantity has changed, be sure to pay attention to whether a percent increase or a growth factor is being discussed.

2 EXPONENTIAL GROWTH PRACTICE 2. The population of Winnemucca, Nevada, can be modeled by P = 6191(1.04) where t is the number of years since What was the population in 1990? By what percent did the population increase each year? 3. Population In 1980 about 2,180,000 U.S. workers worked at home. During the next ten years, the number of workers working at home increased 5% per year. a. Write a model giving the number w (in millions) of workers working at home t years after b. About how many workers were there in 1990? 4. In 1990 the cost of tuition at a state university was $4300. During the next 8 years, the tuition rose 4% each year. c. Write a model that gives the tuition y (in dollars) t years after d. About how much was the tuition in 1994?

3 E XPONENTIAL D ECAY Exponential Decay Formula: a = r = t = 1. You buy a new car for $24,000. The value y of the car decreases by 16% each year. a. Write an exponential decay model for the value of the car. b. Use the model to estimate the value after 2 years. Decay Factor vs. Percent Decrease In the previous example, notice that the percent decrease, 16%, tells you how much value the car loses from one year to the next. The decay factor, 0.84, tells you what fraction of the car s value remains from one year to the next. The closer the percent decrease for some quantity is to 0%, the more the quantity is conserved or retained over time. The closer the percent decrease is to 100%, the more the quantity is used or lost over time.

4 EXPONENTIAL DECAY PRACTICE 2. There are 40,000 homes in your city. Each year 10% of the homes are expected to disconnect from septic systems and connect to the sewer system. a. Write an exponential decay model for the number of homes that still use septic systems. b. Use the model to estimate the number of homes using septic systems after 5 years. 3. A new car costs $23,000. The value decreases by 15% each year. a. Write an exponential decay model for the car s value. b. Use the model to estimate the value after 3 years. 4. You have bought a new car for $24,000. The value of y of the car decreases by 13% each year. a. Write an exponential decay model for the value of the car. b. Use the model to estimate the value of the car after two years. 5. State whether ƒ(x) is an exponential growth or exponential decay function. a. f x = 5 c. f x = 2 b. f x = d. f x = 2.4 3

5 C OMPOUNDED I NTEREST Exponential growth functions are used in real- life situations involving compound interest. Compound interest is interest paid on the initial investment, called the principal, and on previously earned interest. (Interest paid only on the principal is called simple interest.) Although interest earned is expressed as an annual percent, the interest is usually compounded more frequently than once per year. Therefore, the formula y = a(1 + r) must be modified for compound interest problems. Compound Interest Formula = P = r = n = t = Phrase # of times per year Annually Quarterly Monthly Weekly Daily 1. You deposit $1500 in an account that pays 3.5% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency. a. Quarterly b. Monthly

6 2. You deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency. a. Annually b. Quarterly c. Daily 3. You deposit $500 in an account that pays 3% annual interest. Find the balance after 2 years if the interest is compounded with the given frequency. a. Annually b. Quarterly c. Daily 4. You deposit $1500 in an account that pays 6% annual interest. Find the balance after 1 year if the interest is compounded d. Monthly e. Weekly 5. You deposit $1500 in an account that pays 6% annual interest. Find the balance after 1 year if the interest is compounded f. Annually g. Quarterly 6. You deposit $2000 to an account that pays 8% annual interest. How much more does the account earn if the interest is compounded monthly rather than annually?

7 C ONTINUOUSLY C OMPOUNDED I NTEREST Continuous Compounding Interest Formula = P = e = r = t = 1. You deposit $1400 in an account that pays 4% annual interest compounded continuously. What is the balance after 1 year? CONTINUOUSLY COMPOUNDED INTEREST PRACTICE 2. You deposit $800 in an account that pays 9% annual interest compounded continuously. What is the balance after 1 year? 3. You deposit $1500 in an account that pays 7.5% annual interest compounded continuously. What is the balance after 1 year? 4. The atmospheric pressure P (in pounds per square inch) of an object d miles above sea level can be modeled by P = 14.7e.". How much pressure per square inch would you experience at the summit of Mount Washington, 6288 feet above sea level?

Honors Pre-Calculus 3.5 D1 Worksheet Name Exponential Growth and Decay

Honors Pre-Calculus 3.5 D1 Worksheet Name Exponential Growth and Decay Exponential Growth: Exponential Decay: Compound Interest: Compound Interest Continuously: 1. The value in dollars of a car years from