Let's break down each of the scenarios according to the instructions provided.
Scenario 1: Manatee Population
- Part A: Since the manatee population is increasing, this is a growth situation.
- Part B: The rate of growth is provided as 3.1%, or in decimal form, 0.031.
- Part C: The function can be written as: \[ f(x) = 1014(1 + 0.031)^{t} = 1014(1.031)^{t} \]
Scenario 2: Caffeine Metabolism
- Part A: Since the amount of caffeine is decreasing due to metabolism, this is a decay situation.
- Part B: The rate of decay is 3%, or in decimal form, 0.03.
- Part C: The function can be written as: \[ f(x) = 25(1 - 0.03)^{t} = 25(0.97)^{t} \]
Scenario 3: High School Population
- Part A: Since the population of the high school is increasing, this is a growth situation.
- Part B: The rate of growth is 1.18%, or in decimal form, 0.0118.
- Part C: The function can be written as: \[ f(x) = 314(1 + 0.0118)^{t} = 314(1.0118)^{t} \]
Scenario 4: Depreciation of Toyota Prius
- Part A: Since the value of the car is decreasing, this is a decay situation.
- Part B: The rate of decay is 10.5%, or in decimal form, 0.105.
- Part C: The function can be written as: \[ f(x) = 44205(1 - 0.105)^{t} = 44205(0.895)^{t} \]
Summary Table:
| Scenario | Decay or Growth? | r-value | Function | |-------------------------------------------------------------------------------------------------------|-------------------|---------|---------------------------| | South Carolina has found that the manatee population is increasing. | growth | 0.031 | \( f(x) = 1014(1.031)^{t} \) | | A typical cup of tea contains about 25 milligrams (mg) of caffeine. Every hour, approximately 3% of the amount of caffeine in the body is metabolized and eliminated. | decay | 0.03 | \( f(x) = 25(0.97)^{t} \) | | In 2024, the population of the local high school is 314 students with an annual rate of increase of 1.18%. | growth | 0.0118 | \( f(x) = 314(1.0118)^{t} \) | | The value of a 2024 Toyota Prius was $44,205 when it went on the market. It depreciates at a rate of 10.5% per year. | decay | 0.105 | \( f(x) = 44205(0.895)^{t} \) |