Question
10. Exponential Functions in Context(1 point)
Complete the table below.
Recall that the equations for Exponential Growth is f(x) = a(1+r)t
and the Exponential Rate of Decay is f(x) = a(1−r)t
Part A: Determine if the situation is a rate of decay or growth. Write "decay" or "growth"
Part B: Determine the rate of decay or growth (r-value).
Part C: Determine the function of the scenario. Do not include commas in your equation. Simplify your function.
Scenario Decay or Growth? r-value Function
South Carolina has found that the manatee population is increasing.
In 2024, there were 1,014 manatees with a rate of increase of 3.1% per year.
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.
f(x) =
In 2024, the population of the local high school is
314 students with an annual rate of increase of 1.18%.
f(x) =
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
f(x) =
Complete the table below.
Recall that the equations for Exponential Growth is f(x) = a(1+r)t
and the Exponential Rate of Decay is f(x) = a(1−r)t
Part A: Determine if the situation is a rate of decay or growth. Write "decay" or "growth"
Part B: Determine the rate of decay or growth (r-value).
Part C: Determine the function of the scenario. Do not include commas in your equation. Simplify your function.
Scenario Decay or Growth? r-value Function
South Carolina has found that the manatee population is increasing.
In 2024, there were 1,014 manatees with a rate of increase of 3.1% per year.
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.
f(x) =
In 2024, the population of the local high school is
314 students with an annual rate of increase of 1.18%.
f(x) =
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
f(x) =
Answers
GPT-4o mini
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 | \( f(x) = 44205(0.895)^{t} \) |
### 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 | \( f(x) = 44205(0.895)^{t} \) |