Question

The population of Franklin has been decreasing exponentially since the major employer moved out of town. After 3 years the population was 70,496 and after 4 years it was 62,742. Write an equation for, the population of Franklin, years after the employer left town.

Answer 1

The population of Franklin, years after the employer left town, can be modeled by the exponential decay equation
`\(P(t) = P_0 * e^(kt)\),`where P
`^(^t^)`is the population after t years,
`\(P_0\)` is the initial population, and k is the decay constant. For this specific scenario, the equation is
`\(P(t) = 70,496 * e^(kt)\).`

Step-by-step explanation:

The given information states that after 3 years, the population was 70,496, and after 4 years, it was 62,742. We can use these data points to determine the decay constant k. When t = 3, P(3) = 70,496, and when t = 4, P(4) = 62,742.

Substitute these values into the equation:

`\[70,496 = P_0 * e^(3k)\]`

`\[62,742 = P_0 * e^(4k)\]`

Now, divide the two equations to eliminate
`\(P_0\):`

`\[(70,496)/(62,742) = (e^(3k))/(e^(4k))\]`

Simplify the expression on the right side:

`\[(70,496)/(62,742) = e^(-k)\]`

Now, solve for k:

`\[e^(-k) = (70,496)/(62,742)\]`

Take the natural logarithm (ln) of both sides:

`\[-k = \ln\left((70,496)/(62,742)\right)\]`

Finally, solve for k:

`\[k = -\ln\left((70,496)/(62,742)\right)\]`

Now that we have k, we can use it in the exponential decay equation
`\(P(t) = 70,496 * e^(kt)\)` to model the population of Franklin for any given time t after the employer left town.