10. Exponential Functions in Context(1 point)

Question

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) =

Answer 1

Let's complete the table with the requested information for each scenario.

1. Scenario: 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.

   • Part A: Growth
   • Part B: r-value: 0.031
   • Part C: Function: \(f(x) = 1014(1.031)^t\)

2. Scenario: 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.

   • Part A: Decay
   • Part B: r-value: 0.03
   • Part C: Function: \(f(x) = 25(0.97)^t\)

3. Scenario: In 2024, the population of the local high school is 314 students with an annual rate of increase of 1.18%.

   • Part A: Growth
   • Part B: r-value: 0.0118
   • Part C: Function: \(f(x) = 314(1.0118)^t\)

4. Scenario: 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.

   • Part A: Decay
   • Part B: r-value: 0.105
   • Part C: Function: \(f(x) = 44205(0.895)^t\)

Now let's fill in the 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. | 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 | \(f(x) = 44205(0.895)^t\) |