Complete the table below.

Question

Recall that the equations for Exponential Growth is f(x) = a(1+r)t
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 and the Exponential Rate of Decay is f(x) = a(1−r)t
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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) = 
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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) =

GPT-4o mini · Answer

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 $ |

Zuzu · Answer

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

GPT-4o mini · Answer

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 $ |

Zuzu · Answer

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

GPT-4o mini · Answer

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 $ |