The population of Cary in 1980 was 21763. In 1987, the population had grown to 39387. Using the uninhibited growth model, predict the population of Cary for the year 2006.
Understand the Problem
The question is asking us to predict the population of Cary for the year 2006 using the uninhibited growth model based on population data from 1980 and 1987.
Answer
$$ P(2006) = x e^{19r} $$
Answer for screen readers
The final predicted population for Cary in 2006 will depend on the actual values of ( P_0 ) and ( P_1 ).
If we let ( P_0 = x ) (population in 1980) and ( P_1 = y ) (population in 1987), the general formula for the predicted population becomes:
$$ P(2006) = x e^{19r} $$
where you would need to compute ( r ) first.
Steps to Solve
- Identify the given data points
We were provided with population data for the years 1980 and 1987. Let's denote these as:
- Year 1980: ( P_0 )
- Year 1987: ( P_1 )
Assuming we have the actual population numbers, substitute those values into our equations.
- Calculate the growth rate
To find the growth rate ( r ), we can use the formula for exponential growth:
$$ r = \frac{P_1}{P_0} - 1 $$
However, since this represents year intervals, we also need to account for the time period. Here, ( t = 1987 - 1980 = 7 ) years. We can express the population at the later year as:
$$ P(t) = P_0 e^{rt} $$
where ( P(t) ) is the population at time ( t ).
- Express the population model
Rearranging the exponential growth formula gives:
$$ P_1 = P_0 e^{7r} $$
This equation helps us calculate ( r ) if we know ( P_0 ) and ( P_1 ).
- Predict the population for 2006
To predict the population for the year 2006, which is 19 years after 1987, we can substitute into the population model:
$$ P(2006) = P_0 e^{19r} $$
- Plug in the values and solve
Using the value of ( r ) found previously, calculate ( P(2006) ) by substituting ( P_0 ) and ( r ) back into the equation from the previous step.
The final predicted population for Cary in 2006 will depend on the actual values of ( P_0 ) and ( P_1 ).
If we let ( P_0 = x ) (population in 1980) and ( P_1 = y ) (population in 1987), the general formula for the predicted population becomes:
$$ P(2006) = x e^{19r} $$
where you would need to compute ( r ) first.
More Information
The exponential growth model assumes that the population will continue to grow at a constant rate based on previous values. This model can help forecast future populations in various fields like ecology and urban planning.
Tips
- Not accounting for the correct time interval when calculating the growth rate.
- Assuming linear growth rather than exponential growth, which would lead to inaccurate predictions.
