Question

in a census, rose hills had a population of 400. after 13 years, the population grew to 873. assuming the population of rose hills grows at a continuous rate, write an exponential model for this situation in terms of t, the number of years passed. round to two decimal places, when necessary.

Explanation:

Step1: Recall continuous growth formula

The general model for continuous exponential growth is $P(t) = P_0 e^{rt}$, where $P_0$ is the initial population, $r$ is the continuous growth rate, and $t$ is time in years.

Step2: Identify known values

We know $P_0 = 400$, $t = 13$, and $P(13) = 873$. Substitute these into the formula:
$873 = 400 e^{13r}$

Step3: Isolate the exponential term

Divide both sides by 400:
$\frac{873}{400} = e^{13r}$
$2.1825 = e^{13r}$

Step4: Solve for $r$ using natural log

Take the natural logarithm of both sides:
$\ln(2.1825) = 13r$
Calculate $\ln(2.1825) \approx 0.779$, then:
$r = \frac{0.779}{13} \approx 0.06$

Step5: Write the final model

Substitute $P_0 = 400$ and $r \approx 0.06$ back into the original formula.

Answer:

$P(t) = 400e^{0.06t}$