Question

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 2001.

Answer 1

If the population of cary in 1980 was 21763. The predicted population of Cary for the year 2001 using the uninhibited growth model is 123,766.

What is the Predicted population?

Growth rate:

Growth rate = (final population - initial population) / initial population.

Growth rate = (39387 - 21763) / 21763

Growth rate = 0.808

Predicted population = initial population * (1 + growth rate)^n

where:

n= number of years between the initial and predicted populations.

Number of years between 1980 and 2001 is 21

Predicted population =
`21763 * (1 + 0.808)^2^1`

Predicted population ≈ 21763 * 5.687

Predicted population ≈ 123,766

Therefore, the predicted population of Cary for the year 2001 using the uninhibited growth model is approximately 123,766.

Answer 2

Using the uninhibited growth model, the predicted population of Cary for 2001 is around 160,005.9, calculated by extrapolating growth from 1980 to 1987 and applying it over 21 years.

1. Finding the growth rate r:

P₀ = 21763

P(7) = 39387

t = 7

Using the formula
`\( P(t) = P_0 * e^(rt) \)` and the provided data:

`\[ 39387 = 21763 * e^(7r) \]`

To find r, divide both sides by 21763:

`\[ (39387)/(21763) = e^(7r) \]`

Taking the natural logarithm of both sides:

`\[ \ln\left((39387)/(21763)\right) = 7r \]`

Solving for r:

`\[ r = (\ln\left((39387)/(21763)\right))/(7) \]`

`\[ r \approx 0.095 \]`

2. Predicting the population in 2001:

`\[ t = 2001 - 1980 = 21 \]`

Using the growth rate r we calculated, let's predict the population in 2001:

`\[ P(21) = 21763 * e^(21r) \]`

`\[ P(21) \approx 21763 * e^(21 * 0.095) \]`

`\[ P(21) \approx 21763 * e^(1.995) \]`

`\[ P(21) \approx 21763 * 7.3522 \]`

`\[ P(21) \approx 160005.9 \]`

Therefore, using the uninhibited growth model, the predicted population of Cary for the year 2001 is approximately 160005.9.