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) Picture the equation. It must decrease at an unknown rate, then flatten out as it nears 60K (asymptote). This would be in the family of functions of 1/x because 1/x is the most common function to have asymptotes, which Google seems to call power / reciprocal / rational / inverse function.
2) o.o What class is this for? That's definitely not enough information for any sort of regression model, so the only thing I could think of is guess and check?
1. The kind of function that would best model the population over time in this scenario is an exponential decay function. This is because the population is decreasing over time.
2. To write an equation that models the changing population over time, we can use the general form of an exponential decay function:
P(t) = P₀ * e^(kt)
Where:
- P(t) is the population at time t
- P₀ is the initial population
- e is Euler's number (approximately 2.71828)
- k is the decay rate constant
To find the specific equation for this scenario, we need to determine the values of P₀ and k.
Given that in 1998 the population was 72,000, we can use this as our initial population:
P₀ = 72,000
To find k, we can use the fact that the population dropped to 70,379 in 2000. This means the time difference is 2000 - 1998 = 2 years. Using the formula for exponential decay, we can set up the following equation:
70,379 = 72,000 * e^(2k)
Solving this equation will give us the value of k, which we can then substitute back into the exponential decay function to obtain the final equation.
To determine the kind of function that would best model the population over time, we need to examine the given information.
1. The population in 1998 is 72,000 and in 2000 it is 70,379. We can see that the population is decreasing.
2. City officials expect the population to eventually level off at 60,000.
Based on these details, the best function to model this situation would be an exponential decay function. An exponential decay function is commonly used when we have a decreasing quantity over time.
Now, let's write an equation to model the changing population over time.
An exponential decay function can be written in the form: P(t) = P0 * e^(kt)
Where:
- P(t) represents the population at time t
- P0 represents the initial population
- k represents the decay constant
Since we are given the initial population in 1998 as 72,000, we can use this value as P0. The population in 2000 is 70,379, so we have P(2) = 70,379.
Now, let's solve for the decay constant (k) using the provided information.
P(2) = P0 * e^(2k)
70,379 = 72,000 * e^(2k)
Divide both sides of the equation by 72,000:
70,379 / 72,000 = e^(2k)
Take the natural logarithm (ln) of both sides to isolate the exponent:
ln(70,379 / 72,000) = 2k
Now, divide both sides by 2 to solve for k:
k = (1/2) * ln(70,379 / 72,000)
We can then substitute this value of k into our original equation to model the changing population over time:
P(t) = 72,000 * e^((1/2) * ln(70,379 / 72,000))