Exponential Growth and Decay Notes & Practice Problems

Exponential Growth: The process by which a quantity increases rapidly over time, often represented by the equation y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential Decay: The process by which a quantity decreases rapidly over time. This is often represented by the equation y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time. In this case, the quantity reduces as time progresses, contrasting with exponential growth, which involves an increase in quantity.

Example problems:

Problem 1: Your starting salary at a new company is $26,600, and it increases 2.5% per year. Write a function to represent the situation. What will be your new salary in 4 years? Round your answer to the nearest dollar.

Problem 2: A computer that costs $2300 increases at the rate of 10% per year. Write a function that models the value, and find the value of the computer in 5 years. Round to the nearest cent.

Problem 3: A small town in 2010 had a population of approximately 10,000 people. The annual rate of increase until 2025 was about 4.52%. Write a function that represents the situation. What is the population of the town in 2025?

Problem 4: A large city in 2018 has a population of 200,000 people. The population decreases by around 6.8% per year. Write a function and find the population count by 2023. Round to the nearest person.

Problem 5: On Monday morning around 8:00 am, a virus spreads across Sweetwater High School with a initial infection count of 15 people (both students and staff). The school closes as the number of infections increases by 36% per hour. Write an exponential function that represents this problem. Out of 2,000 students, how many people are infected by 3:00 pm? Round to the nearest person.

Answers:

Answer to problem 1:

The growth factor is .025.

Function: 1 + .025 = 1.025.

Your equation should be: y = $26,600 (1.025)^4

Your answer: $29361

Answer to problem 2:

Your growth factor is .10.

Function: 1 + .10 = 1.10.

Your equation should be: y = $2300 (1.10)^6

Your answer: $3704.17

Answer to problem 3:

The growth factor is .0452.

Function: 1 + .0452 = 1.0452

Your equation: y = 10000(1.0452)^15

Your answer: 19408 people.

Answer to problem 4:

The decay factor is .068.

Function: 1 - .068 = .932

Your equation: y = 200000(.932)^5

Your answer: 140,640 people.

Answer to problem 5:

The growth factor is .36.

Function: 1 + .36 = 1.36

Your equation: 15(1.36)^7

Your answer: 129 people.