Question

exponential function that represents the population growth of a town for a current population of 5,400, an initial population of 5,100, and a time period of 10 years.

rowth rate, r, for the equation written in part a. round to the nearest thousandth. show your work.

Explanation:

Step1: Recall exponential growth formula

The general formula for exponential growth is $P(t) = P_0(1 + r)^t$, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $r$ is the growth rate, and $t$ is the time. Here, $P_0 = 5100$, $P(10)=5400$, and $t = 10$.

Step2: Substitute values into the formula

Substitute $P(10) = 5400$, $P_0 = 5100$, and $t = 10$ into the formula: $5400 = 5100(1 + r)^{10}$.

Step3: Solve for $(1 + r)$

First, divide both sides by 5100: $\frac{5400}{5100}=(1 + r)^{10}$. Simplify $\frac{5400}{5100}=\frac{18}{17}\approx1.058824$. So, $1.058824=(1 + r)^{10}$.

Step4: Take the 10th root of both sides

Take the 10th root of both sides (or raise both sides to the power of $\frac{1}{10}$): $1 + r = 1.058824^{\frac{1}{10}}$. Calculate $1.058824^{\frac{1}{10}}$: using a calculator, $1.058824^{\frac{1}{10}}\approx1.00566$.

Step5: Solve for $r$

Subtract 1 from both sides: $r = 1.00566 - 1 = 0.00566$. Rounding to the nearest thousandth, $r\approx0.006$.

Answer:

The growth rate $r$ is approximately $\boldsymbol{0.006}$.