# Suppose that a population can be modeled with exponential growth. Initially, the population...

## Question:

Suppose that a population can be modeled with exponential growth. Initially, the population consists of {eq}100 {/eq} individuals. After {eq}t=25 {/eq} years, the population consists of {eq}400 {/eq} individuals. Determine the model for the population {eq}P {/eq} as a function of time {eq}t {/eq} elapsed: {eq}P(t)= {/eq}

## Exponential Functions:

From the mathematical representation shown {eq}P(t) = P(0)e^{rt} {/eq}, exponential functions show the behavior of a population as time progresses. From the function, {eq}P(t) {/eq} serves to show the population at the specific time {eq}t {/eq}, {eq}P(0) {/eq} is the initial population, {eq}r {/eq} is the rate of growth, and {eq}t {/eq} is the specified time point.

Given: {eq}P(0) = 100 \\ P(25) = 400 {/eq}

To determine the exponential model for the population function {eq}P(t) {/eq}, the exponential growth equation ({eq}P(t) = P(0)e^{rt} {/eq}) will be applied in this case to solve for the unknown value of {eq}r {/eq}.

{eq}\begin{align*} P(25) = P(0)e^{r(25)} &\Rightarrow 400 = 100e^{25r} \\ &\Rightarrow 400\div 100 = 100e^{25r}\div 100 \\ &\Rightarrow 4 = e^{25r} \\ &\Rightarrow \ln(4) = \ln(e^{25r}) \\ &\Rightarrow \ln(4) = 25r \\ &\Rightarrow \ln(4)\div 25 = 25r\div 25 \\ &\Rightarrow r = 0.0554517744 \end{align*} {/eq}

Therefore, at any time point {eq}t {/eq}, the population function can be modeled by the equation {eq}P(t) = 100e^{0.0554517744t} {/eq}. 