Unit 8: Exponential and Logarithmic Modeling
Turning growth, decay, and compounding money into exponential equations you can actually solve for time or amount.
5 topics · 18 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Exponential Growth and Decay Models
- How it worksHow does a percent growth or decay rate turn into the base of an exponential model?
- Worked exampleFind the population in 10 years for a town of 12000 people growing 3 percent per year using an exponential model.
- Common mistakeWhy doesn't a 3 percent growth rate mean the population grows by the same number of people every year?
Your turn1 of 2
Write the population P(t) in a town of 36339 people growing 7 percent per year, as a function of the number of years t.Half-Life and Radioactive Decay
- How it worksHow does the half-life of a substance become the base of its exponential decay model?
- Worked exampleFind how much of a 200 gram sample remains after 3 half-lives if the half-life is 5 years.
- Common mistakeWhy doesn't a substance with a 10 year half-life lose all its material in 20 years?
- Going furtherWhat changes about the decay model's formula when a problem gives a yearly decay rate instead of a half-life?
Your turn1 of 2
Write the amount M(n) remaining of a 363 gram sample after n half-lives, as a function of n.Compound Interest
- How it worksHow does the compound interest formula multiply your balance by the same growth factor over and over?
- Worked exampleFind the balance after 6 years on a 3000 dollar investment at 5 percent annual interest compounded monthly.
- Common mistakeWhy doesn't doubling the compounding frequency come anywhere close to doubling your final balance?
- Going furtherWhat happens to the compound interest formula's result as the number of compounding periods per year grows very large?
Your turn1 of 2
Write the balance A(t) after t years on a 7116 dollar investment at 7 percent annual interest compounded monthly, as a function of t.Continuous Compounding and the Number e
- How it worksHow does compounding infinitely often turn the compound interest formula into one built on the number e?
- Worked exampleFind how much a 1200 dollar deposit grows to after 8 years at 6 percent interest compounded continuously.
- Common mistakeWhy isn't e just an approximation like 2.7, and what makes it an exact, specific number?
Your turn1 of 2
Write how much a 7116 dollar deposit grows to after t years at 7 percent interest compounded continuously, as a function of t.Fitting an Exponential Model to Data
- How it worksHow do you find an exponential model's growth factor from two data points a known number of steps apart?
- Worked exampleFind an exponential model given that a culture had 500 bacteria at hour 0 and 4000 bacteria at hour 3.
- Common mistakeWhy doesn't a constant difference between data points ever point to an exponential model?
- Ties togetherHow does finding a growth factor from two data points use the same idea as the rate in a half-life or interest formula?
Your turn1 of 2
A culture had 727 bacteria at hour 0 and 5894 bacteria at hour 5. Write the exponential model N(x) fitted to these two data points, as a function of x.