Answer :
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.
### 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.