Question
A small town’s population, , can be modeled by the equation , where is the time in years.
Indicate whether each statement is true or false.
True False
The initial population of the town at year is people.
The population of the town at years is people.
The rate of population decay each year is .
The rate of population growth each year is .
Indicate whether each statement is true or false.
True False
The initial population of the town at year is people.
The population of the town at years is people.
The rate of population decay each year is .
The rate of population growth each year is .
Answers
GPT-4o mini
To accurately assess the truth of each statement, we need the specific equation for the population of the town. Since you haven't provided it, I will outline how you might evaluate those statements based on a general form of population equations.
Let's say the population, \( P(t) \), is modeled by an equation depending on the context. Common forms include:
1. **Exponential Growth:** \( P(t) = P_0 e^{kt} \)
2. **Exponential Decay:** \( P(t) = P_0 e^{-kt} \)
3. **Linear Growth/Decay:** \( P(t) = P_0 + rt \)
Where:
- \( P_0 \) is the initial population,
- \( k \) is the rate of growth (when \( k > 0 \)), or the rate of decay (when \( k < 0 \)),
- \( r \) is a constant indicating the rate per year.
Now let's evaluate the statements:
1. **The initial population of the town at year 0 is \( P_0 \) people.**
- If the equation is evaluated at \( t = 0 \), that generally yields the initial population, \( P(0) = P_0 \). Thus, this statement is **True**.
2. **The population of the town at \( t = t_0 \) years is \( P(t_0) \) people.**
- This statement depends directly on evaluating \( P(t) \) at \( t = t_0 \), which gives you the population at that time. If \( t_0 \) is specified in the model, then the statement is also **True**.
3. **The rate of population decay each year is \( -k \).**
- If it’s a decay model like \( P(t) = P_0 e^{-kt} \), then the decay rate indeed would be parameterized by \( k \). However, this statement requires the specific form of the model to determine if \( -k \) is correct. If the decay is exponential, this statement is **True**.
4. **The rate of population growth each year is \( k \).**
- Similarly, if the model indicates growth (i.e., \( k > 0 \)), then this statement would hold true. If it's a decay model or if \( k \) is negative (indicating that the population is falling), then this statement would be **False**.
To conclude, the truthfulness of these statements relies heavily on the specifics of the population equation you are using. Please provide the model so I can give tailored answers according to it.
Let's say the population, \( P(t) \), is modeled by an equation depending on the context. Common forms include:
1. **Exponential Growth:** \( P(t) = P_0 e^{kt} \)
2. **Exponential Decay:** \( P(t) = P_0 e^{-kt} \)
3. **Linear Growth/Decay:** \( P(t) = P_0 + rt \)
Where:
- \( P_0 \) is the initial population,
- \( k \) is the rate of growth (when \( k > 0 \)), or the rate of decay (when \( k < 0 \)),
- \( r \) is a constant indicating the rate per year.
Now let's evaluate the statements:
1. **The initial population of the town at year 0 is \( P_0 \) people.**
- If the equation is evaluated at \( t = 0 \), that generally yields the initial population, \( P(0) = P_0 \). Thus, this statement is **True**.
2. **The population of the town at \( t = t_0 \) years is \( P(t_0) \) people.**
- This statement depends directly on evaluating \( P(t) \) at \( t = t_0 \), which gives you the population at that time. If \( t_0 \) is specified in the model, then the statement is also **True**.
3. **The rate of population decay each year is \( -k \).**
- If it’s a decay model like \( P(t) = P_0 e^{-kt} \), then the decay rate indeed would be parameterized by \( k \). However, this statement requires the specific form of the model to determine if \( -k \) is correct. If the decay is exponential, this statement is **True**.
4. **The rate of population growth each year is \( k \).**
- Similarly, if the model indicates growth (i.e., \( k > 0 \)), then this statement would hold true. If it's a decay model or if \( k \) is negative (indicating that the population is falling), then this statement would be **False**.
To conclude, the truthfulness of these statements relies heavily on the specifics of the population equation you are using. Please provide the model so I can give tailored answers according to it.