In 1998, Terre Haute, Indiana had a population of 72,000 people. In 2000, the population had dropped to 70,379. City officials expect the population to level off eventually at 60,000.

1. What kind of function would best model the population over time ?

2. Write an equation that would model the changing population over time?

1. Based on the given information, a linear function would be the most appropriate to model the population over time. This is because the population is decreasing at a consistent rate over a specific period.

2. To write an equation that models the changing population over time, we can use the point-slope form of a linear equation. Let's use the year as the independent variable (x) and the population as the dependent variable (y).

We have two data points: (1998, 72,000) and (2000, 70,379). Let's use these points to find the slope:

Slope (m) = (change in y) / (change in x)
= (70,379 - 72,000) / (2000 - 1998)
= -821 / 2
= -410.5

Now that we have the slope, we can use the point-slope form with the first data point:

y - y1 = m(x - x1)

Substituting (x1, y1) = (1998, 72,000) and the slope (m = -410.5), we get:

y - 72,000 = -410.5(x - 1998)

Simplifying this equation gives us the final equation that models the population over time:

y = -410.5x + 821601.5

To determine the best type of function that would model the population over time, we should consider the characteristics of the given data. In this case, we have two data points: the population in 1998 and the population in 2000.

1. Based on the given data, the population is decreasing over time. Therefore, a decreasing function would be appropriate. Additionally, since the city officials expect the population to level off eventually, an exponential decay function would be suitable.

2. The equation for exponential decay is generally given as:

P(t) = P0 * e^(kt)

where:
- P(t) represents the population at time t
- P0 is the initial population
- e is the base of the natural logarithm
- k is the decay constant

To find the specific equation that models the changing population in Terre Haute over time, we need to determine the values of P0, k, and t.

Given:
- The population in 1998: P(0) = 72,000
- The population in 2000: P(2) = 70,379

To find 'k', we can use the formula:

k = ln(P(2)/P(0))/2

Substituting the values:

k = ln(70,379/72,000)/2

Using a calculator, we can find:

k ≈ -0.0092

Now, we can substitute the values of P0 = 72,000 and k ≈ -0.0092 into the equation:

P(t) = 72,000 * e^(-0.0092t)

This equation would model the changing population over time in Terre Haute, Indiana, where P(t) represents the population at time t.

some kind of exponential would do it. So, with t as time since 1998,

f(0) = 72000
f(2) = 70379
f(∞) = 60000

f(t) = 60000 + 12000 e^-kt

now we need k so that f(2) = 70379

60000 + 12000 e^-2k = 10379
k = 0.073

So,

f(t) = 60000 + 12000 e^(-.073t)