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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) =

1 answer

Let's complete the table according to the scenarios you provided.

  1. South Carolina has found that the manatee population is increasing.

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

  2. 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 \( f(x) = 25(1-0.03)^t = 25(0.97)^t \)

  3. 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. 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 \( f(x) = 44205(1-0.105)^t = 44205(0.895)^t \)

Here’s the completed 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 | \( f(x) = 44205(0.895)^t \) |