Question
Complete the table below.
Recall that the equations for Exponential Growth is f(x) = a(1+r)t
𝑓
(
𝑥
)
=
𝑎
(
1
+
𝑟
)
𝑡
and the Exponential Rate of Decay is f(x) = a(1−r)t
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) =
𝑓
(
𝑥
)
=
1014(1.031)t
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) =
Recall that the equations for Exponential Growth is f(x) = a(1+r)t
𝑓
(
𝑥
)
=
𝑎
(
1
+
𝑟
)
𝑡
and the Exponential Rate of Decay is f(x) = a(1−r)t
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) =
𝑓
(
𝑥
)
=
1014(1.031)t
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) =
Answers
GPT-4o mini
Let's complete the table for the scenario involving caffeine metabolism.
### Part A: Determine if the situation is a rate of decay or growth.
This scenario describes a loss of caffeine in the body over time, so it is a "decay."
### Part B: Determine the rate of decay or growth (r-value).
The caffeine is metabolized at a rate of 3%, which means the rate of decay \( r = 0.03 \).
### Part C: Determine the function of the scenario.
The initial amount of caffeine is 25 mg, and since it is decaying, we will use the decay formula:
\[ f(x) = a(1 - r)^t \]
Substituting the values we have:
- \( a = 25 \) mg (initial caffeine),
- \( r = 0.03 \).
So, the function can be expressed as:
\[ f(x) = 25(1 - 0.03)^t \]
\[ f(x) = 25(0.97)^t \]
Putting it all together in the table format:
| Scenario | Decay or Growth? | r-value | Function |
|----------------------------------------------------------------------------------------------------|-------------------|---------|------------------------------|
| 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 | \( f(x) = 25(0.97)^t \) |
### Part A: Determine if the situation is a rate of decay or growth.
This scenario describes a loss of caffeine in the body over time, so it is a "decay."
### Part B: Determine the rate of decay or growth (r-value).
The caffeine is metabolized at a rate of 3%, which means the rate of decay \( r = 0.03 \).
### Part C: Determine the function of the scenario.
The initial amount of caffeine is 25 mg, and since it is decaying, we will use the decay formula:
\[ f(x) = a(1 - r)^t \]
Substituting the values we have:
- \( a = 25 \) mg (initial caffeine),
- \( r = 0.03 \).
So, the function can be expressed as:
\[ f(x) = 25(1 - 0.03)^t \]
\[ f(x) = 25(0.97)^t \]
Putting it all together in the table format:
| Scenario | Decay or Growth? | r-value | Function |
|----------------------------------------------------------------------------------------------------|-------------------|---------|------------------------------|
| 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 | \( f(x) = 25(0.97)^t \) |
Zuzu
In 2024, the population of the local high school is
314 students with an annual rate of increase of 1.18%.
314 students with an annual rate of increase of 1.18%.
GPT-4o mini
Let's analyze the new scenario involving the population of the local high school.
### Part A: Determine if the situation is a rate of decay or growth.
Since the population is increasing, this is a "growth."
### Part B: Determine the rate of decay or growth (r-value).
The annual rate of increase is 1.18%, which means the growth rate \( r \) is:
\[ r = 0.0118 \]
### Part C: Determine the function of the scenario.
The initial population is 314 students. Using the formula for exponential growth:
\[ f(x) = a(1 + r)^t \]
Substituting the values:
- \( a = 314 \) (initial population),
- \( r = 0.0118 \).
So the function can be expressed as:
\[ f(x) = 314(1 + 0.0118)^t \]
\[ f(x) = 314(1.0118)^t \]
Putting it all together, the completed table is as follows:
| Scenario | Decay or Growth? | r-value | Function |
|------------------------------------------------------------------------------------------------|-------------------|---------|--------------------------|
| In 2024, the population of the local high school is 314 students with an annual rate of increase of 1.18%. | growth | 0.0118 | \( f(x) = 314(1.0118)^t \) |
### Part A: Determine if the situation is a rate of decay or growth.
Since the population is increasing, this is a "growth."
### Part B: Determine the rate of decay or growth (r-value).
The annual rate of increase is 1.18%, which means the growth rate \( r \) is:
\[ r = 0.0118 \]
### Part C: Determine the function of the scenario.
The initial population is 314 students. Using the formula for exponential growth:
\[ f(x) = a(1 + r)^t \]
Substituting the values:
- \( a = 314 \) (initial population),
- \( r = 0.0118 \).
So the function can be expressed as:
\[ f(x) = 314(1 + 0.0118)^t \]
\[ f(x) = 314(1.0118)^t \]
Putting it all together, the completed table is as follows:
| Scenario | Decay or Growth? | r-value | Function |
|------------------------------------------------------------------------------------------------|-------------------|---------|--------------------------|
| In 2024, the population of the local high school is 314 students with an annual rate of increase of 1.18%. | growth | 0.0118 | \( f(x) = 314(1.0118)^t \) |
Zuzu
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
market. It depreciates at a rate of 10.5% per year
GPT-4o mini
Let's analyze the scenario involving the depreciation of a 2024 Toyota Prius.
### Part A: Determine if the situation is a rate of decay or growth.
Since the value of the Toyota Prius is decreasing over time due to depreciation, this is a "decay."
### Part B: Determine the rate of decay or growth (r-value).
The depreciation rate is 10.5%, so the decay rate \( r \) is:
\[ r = 0.105 \]
### Part C: Determine the function of the scenario.
The initial value of the Toyota Prius is $44,205. Using the formula for exponential decay:
\[ f(x) = a(1 - r)^t \]
Substituting the values we have:
- \( a = 44205 \) (initial value),
- \( r = 0.105 \).
So, the function can be expressed as:
\[ f(x) = 44205(1 - 0.105)^t \]
\[ f(x) = 44205(0.895)^t \]
Putting it all together in the completed table format:
| Scenario | Decay or Growth? | r-value | Function |
|-------------------------------------------------------------------------------------------------------------|-------------------|---------|-----------------------------|
| 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 | \( f(x) = 44205(0.895)^t \) |
### Part A: Determine if the situation is a rate of decay or growth.
Since the value of the Toyota Prius is decreasing over time due to depreciation, this is a "decay."
### Part B: Determine the rate of decay or growth (r-value).
The depreciation rate is 10.5%, so the decay rate \( r \) is:
\[ r = 0.105 \]
### Part C: Determine the function of the scenario.
The initial value of the Toyota Prius is $44,205. Using the formula for exponential decay:
\[ f(x) = a(1 - r)^t \]
Substituting the values we have:
- \( a = 44205 \) (initial value),
- \( r = 0.105 \).
So, the function can be expressed as:
\[ f(x) = 44205(1 - 0.105)^t \]
\[ f(x) = 44205(0.895)^t \]
Putting it all together in the completed table format:
| Scenario | Decay or Growth? | r-value | Function |
|-------------------------------------------------------------------------------------------------------------|-------------------|---------|-----------------------------|
| 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 | \( f(x) = 44205(0.895)^t \) |