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Let's apply this understanding to situations when we're given the context and function.

1. A new species of fish is released into a lake, and the fish multiply quickly. The growth of their population is modeled by the exponential function [tex]P(t)=5 b^t[/tex], where [tex]t[/tex] is the time in weeks after the release and [tex]b[/tex] is a positive unknown base. According to the model, how many fish of the new species were released initially? [tex]\square[/tex]

2. A study of the annual population of toads in a county park shows the population, [tex]S(t)[/tex], can be represented by the function [tex]S(t)=152(1.045)^t[/tex], where [tex]t[/tex] represents the number of years since the study started. Based on the function, what is the growth rate?
- 4.5%

3. Subscriptions to a popular garden magazine have gone down by a consistent percentage each year and can be modeled by the function [tex]y=24,000(0.89)^t[/tex]. What does the value 24,000 represent in the function? [tex]\square[/tex]

4. Subscriptions to a popular garden magazine have gone down by a consistent percentage each year and can be modeled by the function [tex]y=24,000(0.89)^t[/tex]. What does the value 0.89 represent in the function? [tex]\square[/tex]



Answer :

GinnyAnswer
Let's break down each part of the question one by one, discussing the given mathematical models and interpreting their parameters.

### 1. Initial Number of Fish Released

The growth of the fish population in the lake is given by the function:
[tex]\[ P(t) = 5b^t \][/tex]

Here:
- [tex]\( P(t) \)[/tex] is the population of fish at time [tex]\( t \)[/tex] weeks.
- The number 5 represents the initial number of fish released.
- [tex]\( b \)[/tex] is the unknown base, which indicates the growth rate factor each week.

Parameter of interest: The initial number of fish released, which is when [tex]\( t = 0 \)[/tex].

When [tex]\( t = 0 \)[/tex]:
[tex]\[ P(0) = 5 \cdot b^0 \][/tex]
[tex]\[ P(0) = 5 \cdot 1 \][/tex]
[tex]\[ P(0) = 5 \][/tex]

So, the initial number of fish released into the lake is 5.

### 2. Growth Rate of the Toad Population

The population of toads in the county park is modeled by:
[tex]\[ S(t) = 152(1.045)^t \][/tex]

Here:
- 152 represents the initial population of toads.
- [tex]\( 1.045 \)[/tex] is the growth factor each year.

Parameter of interest: The growth rate of the toad population.

To find the growth rate, observe the base of the exponential function [tex]\( 1.045 \)[/tex].
[tex]\[ 1.045 = 1 + \text{growth rate} \][/tex]

The growth rate is the additional factor [tex]\( 0.045 \)[/tex] above 1, expressed as a percentage:
[tex]\[ 0.045 \times 100 = 4.5 \% \][/tex]

So, the growth rate of the toad population is 4.5%.

### 3. Interpretation of the Value 24,000 for Magazine Subscriptions

The subscriptions to a garden magazine are modeled by:
[tex]\[ y = 24,000(0.89)^t \][/tex]

Here:
- [tex]\( y \)[/tex] represents the number of subscriptions at time [tex]\( t \)[/tex] years.
- 24,000 is a coefficient of the model.

Parameter of interest: The value 24,000.

The value 24,000 represents the initial number of subscriptions at the starting time when [tex]\( t = 0 \)[/tex].

So, 24,000 is the initial number of subscriptions to the magazine.

### 4. Interpretation of the Value 0.89 for Magazine Subscriptions

Continuing with the model:
[tex]\[ y = 24,000(0.89)^t \][/tex]

Parameter of interest: The value 0.89.

This value represents the rate at which the subscriptions are decreasing annually. Specifically, [tex]\( 0.89 \)[/tex] indicates that each year the number of subscriptions is [tex]\( 89\% \)[/tex] of the previous year's subscriptions.

So, 0.89 represents the annual decrease factor for subscriptions to the magazine.

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