Question
The functiongives the approximate number of fish in a large pond after t years. Select all of the true statements.
Responses
There were originally 45 fish in the pond.
There were originally 45 fish in the pond.
There were originally 54 fish in the pond.
There were originally 54 fish in the pond.
The number of fish increases by 9 every year.
The number of fish increases by 9 every year.
The number of fish doubles every year.
The number of fish doubles every year.
The number of fish increases by a factor of 1.2 every year.
Responses
There were originally 45 fish in the pond.
There were originally 45 fish in the pond.
There were originally 54 fish in the pond.
There were originally 54 fish in the pond.
The number of fish increases by 9 every year.
The number of fish increases by 9 every year.
The number of fish doubles every year.
The number of fish doubles every year.
The number of fish increases by a factor of 1.2 every year.
Answers
GPT-4o mini
To determine which statements about the function describing the fish population in the pond are true, we need to analyze the function itself. Since you haven't provided the function or any specific numerical details, I can only guide you on how to assess the statements based on common mathematical models of population growth.
1. **There were originally 45 fish in the pond.** - This statement could only be true if the function indicates that the initial value, when \( t = 0 \), is 45.
2. **There were originally 54 fish in the pond.** - Similar to the first statement, this would be true only if the function's starting value is 54 when \( t = 0 \).
3. **The number of fish increases by 9 every year.** - This indicates a linear growth model. This statement is true if the function has a constant growth rate of 9 fish per year.
4. **The number of fish doubles every year.** - This indicates exponential growth with a doubling behavior. This would be true if the function follows a model like \( N(t) = N_0 \cdot 2^t \).
5. **The number of fish increases by a factor of 1.2 every year.** - This suggests a growth model of the form \( N(t) = N_0 \cdot 1.2^t \). This would be true if the function indicates that the population grows by 20% each year.
To verify the truth of these statements, you would ideally need to refer to the specific mathematical model or function that describes the fish population at time \( t \). If you have that function, you can evaluate its parameters to determine the truth of each statement.
In summary, the truth of each statement depends entirely on the starting point and growth dynamics defined by the function for the fish population.
1. **There were originally 45 fish in the pond.** - This statement could only be true if the function indicates that the initial value, when \( t = 0 \), is 45.
2. **There were originally 54 fish in the pond.** - Similar to the first statement, this would be true only if the function's starting value is 54 when \( t = 0 \).
3. **The number of fish increases by 9 every year.** - This indicates a linear growth model. This statement is true if the function has a constant growth rate of 9 fish per year.
4. **The number of fish doubles every year.** - This indicates exponential growth with a doubling behavior. This would be true if the function follows a model like \( N(t) = N_0 \cdot 2^t \).
5. **The number of fish increases by a factor of 1.2 every year.** - This suggests a growth model of the form \( N(t) = N_0 \cdot 1.2^t \). This would be true if the function indicates that the population grows by 20% each year.
To verify the truth of these statements, you would ideally need to refer to the specific mathematical model or function that describes the fish population at time \( t \). If you have that function, you can evaluate its parameters to determine the truth of each statement.
In summary, the truth of each statement depends entirely on the starting point and growth dynamics defined by the function for the fish population.