Exponential Applications EQ: How do we graph and use exponential functions? M2 Unit 5a: Day 7
Real-World Applications: • Internet traffic growth • The number of microorganisms growing in a culture • The spread of a virus (SARS, West Nile, small pox, etc) • Human population • High profits for a few initial investors in Pyramid schemes or Ponzi schemes • Example on the right: # of cell phone users from 1986-1995 • Compound interest
The Exponential Growth Model • When a real-life quantity is increases by a fixed percent ,r, each year (or other time period), the amount, y, of the quantity after t years can be modeled in this equation: • a = initial amount • r = % increase • (1 + r) = growth factor Example 1: A diamond ring was purchased 20 years ago for $500. The value of the ring increased by 8% each year. What is the value of the ring today?
Example 2: In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville in 1994 if it can be found using the formula a is the initial amount, r is the growth rate, and x is the number of years since 1985?
The Exponential Decay Model • The exponential decay model has the form where y is the quantity after t years, a is the initial amount, r is the percent decrease expressed as a decimal, and the quantity 1 – r is called the decay factor. • Example 3: • Ten grams of Carbon 14 is stored in a container. The amount C (in grams of Carbon 14 present after t years can be modeled by . How much carbon 14 is present after 1000 years?
Example 4: A man purchased a brand new Outlander 800 ATV for $13,000. It depreciates at a rate of 15% per year. What is the value of the Outlander after 5 years?
Compound interest: A = amount of money P = principle or amount initially invested r = compound annual rate n = number of times compounded annually t = time (in years) Example 5: The amount of money, A, accrued at the end of n years when a certain amount P, is invested at a compound annual rate, r, is given by . If a person invests $550 in an account that pays 7% interest compounded annually, find the balance after 5 years.
Compound interest: A = amount of money P = principle or amount initially invested r = compound annual rate n = number of times compounded annually t = time (in years) Example 6: You deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if compounded with the given frequency: (a) Annually (b) Quarterly (c) Daily
Homework • Exponential Handout