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The initial population of a town is 2,600, and it grows with a doubling time of 10 years. What...

Question:

The initial population of a town is 2,600, and it grows with a doubling time of 10 years. What will the population be in 12 years?

Exponential Growth Equations:

When something grows in pattern such that its growth increases with time, we call this an exponential growth. An exponential growth equation takes the form of {eq}P_t = P_0e^{rt} {/eq}.

Answer and Explanation:


The exponential growth equation takes the form:

$$\begin{align} P_t = P_0e^{rt} \end{align} $$

where:

{eq}P_t {/eq} is the population at any Time {eq}t {/eq},

{eq}P_0 {/eq} is the initial population,

{eq}r {/eq} is the growth rate and,

{eq}t {/eq} is the time.

In our question, we have:

{eq}\begin{align} P_0 = 2,600\\[0.3cm] P_10 = 5,200\\[0.3cm] t = 10\\[0.3cm] r = ? \end{align} {/eq}

Therefore:

$$\begin{align} 5,200 = 2,600e^{10r} \end{align} $$

$$\begin{align} 2= e^{10r} \end{align} $$

Introducing natural logarithms on both sides:

$$\begin{align} & \ln 2= \ln e^{10r}\\[0.3cm] & ln 2 = 10r \ln e\\[0.3cm] & \ln 2 = 10r \end{align} $$

Solving for {eq}r {/eq}:

$$\begin{align} & r = \dfrac{\ln 2}{10}\\[0.3cm] r\approx 0.069315 \end{align} $$

Therefore, the growth equation is:

$$\begin{align} P_t = P_0e^{0.069315t} \end{align} $$

Thus, the population after {eq}12 {/eq} years is:

$$\begin{align} & P_{12}= 2,600e^{0.069315\times 12}\\[0.3cm] & P_{12}= 2,600e^{0.83178}\\[0.3cm] & \boxed{\color{blue}{P_{12} \approx 5,973}} \end{align} $$


Learn more about this topic:

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Exponential Growth: Definition & Examples

from High School Algebra I: Help and Review

Chapter 6 / Lesson 10
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