# The formula A = 118e^{0.024t} models the population of a particular city, in thousands, t ...

## Question:

The formula {eq}A = 118e^{0.024t} {/eq} models the population of a particular city, in thousands, {eq}t {/eq} years after 1998. When will the population of the city reach 140 thousand?

## Exponential Growth Function

In mathematical modeling, a function can be used to model quantities that are changing over time. An exponential growth function is used to model quantities that are growing or increasing in number over time. It is used to model population of people in a city or bacteria in a culture medium. It is even used in economics to model simple economic growth.

Suppose the exponential growth function, {eq}A(t) = 118 e^{0.024t} {/eq}, describes the population of a particular city in thousands after the year 1998. If we want to solve the year that the population of the city reaches {eq}A(t) = 140\ \rm{thousand} {/eq}, we plug it into {eq}A(t) {/eq} and solve for t. Note that t here means the number of years after 1998.

Solving for t.

{eq}\displaystyle \begin{align*} A(t) &= 118 e^{0.024t},\ A(t) = 140 \\ 140 &= 118 e^{0.024t} \\ e^{0.024t} &= \frac{140}{118} \end{align*} {/eq}

We can take the natural logarithm of both sides of the equation to remove the exponential and solve for t.

{eq}\displaystyle \begin{align*} \ln \bigg ( e^{0.024t} &= \frac{140}{118} \bigg) \\ 0.024t &= \ln \frac{140}{118}\\ t &= \frac{1}{0.024}\ln \frac{140}{118}\\ t &= \boxed{ 7\ \rm{years}} \end{align*} {/eq}

Therefore, in {eq}\boxed{ \text{ seven years after 1998 or 2005 that city population reaches 140 thousand}} {/eq}.