Alright, let's tackle these two cases and derive the population formulas step-by-step.
### Case (a): The population increases by 90 people per year.
1. Initial Population:
The initial population at time [tex]\( t = 0 \)[/tex] is 2000 people.
2. Annual Increase:
The population increases by 90 people each year.
To set up the formula, we can use a linear growth model because the increase is by a fixed number of people each year. The formula for the population [tex]\( P \)[/tex] as a function of time [tex]\( t \)[/tex] (in years) can be expressed as:
[tex]\[ P = P_0 + \text{increase per year} \times t \][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population (2000 people),
- "increase per year" is 90 people,
- [tex]\( t \)[/tex] is the number of years.
Putting the values into the formula, we get:
[tex]\[ P = 2000 + 90t \][/tex]
So, the population function for case (a) is:
[tex]\[ P = 2000 + 90t \][/tex] people.
### Case (b): The population increases by 1 percent a year.
1. Initial Population:
The initial population at time [tex]\( t = 0 \)[/tex] is 2000 people.
2. Annual Increase Rate:
The population increases by 1% each year. This type of growth is exponential, meaning the population in each subsequent year depends on the population of the previous year multiplied by the growth factor.
To set up the formula, we use an exponential growth model. The general formula for exponential growth is:
[tex]\[ P = P_0 \times (1 + r)^t \][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population (2000 people),
- [tex]\( r \)[/tex] is the annual growth rate (1% or 0.01),
- [tex]\( t \)[/tex] is the number of years.
Substituting the values, we get:
[tex]\[ P = 2000 \times (1 + 0.01)^t \][/tex]
Simplifying the expression inside the parentheses:
[tex]\[ P = 2000 \times (1.01)^t \][/tex]
So, the population function for case (b) is:
[tex]\[ P = 2000 \times 1.01^t \][/tex] people.
In summary, the formulas for the population [tex]\( P \)[/tex] as a function of time [tex]\( t \)[/tex] (in years) are:
- For case (a): [tex]\( P = 2000 + 90t \)[/tex] people.
- For case (b): [tex]\( P = 2000 \times 1.01^t \)[/tex] people.
