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) =

Answer 1

In 2024, the population of the local high school is

314 students with an annual rate of increase of 1.18%.

Answer 2

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

Answer 3

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 | \( f(x) = 25(0.97)^t \) |

Answer 4

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 | \( f(x) = 314(1.0118)^t \) |

Answer 5

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 | \( f(x) = 44205(0.895)^t \) |