Complete question: <span>At the beginning of a population study, a city had 300,000 people. Each year since, the population has grown by 6.8%.
Let t be the number of years since start of the study. Let y be the city's population. Write an exponential function showing the relationship between y and t.
To model this situation we are going to use the standard exponential grow function: </span>
where
![y](https://tex.z-dn.net/?f=y)
is the final population after
![t](https://tex.z-dn.net/?f=t)
years of exponential grow
![a](https://tex.z-dn.net/?f=a)
is the initial population
![b](https://tex.z-dn.net/?f=b)
is the grow rate in decimal form
![t](https://tex.z-dn.net/?f=t)
is the time in years
We know form our problem that the initial population of the city at the beginning of the study was 300,000 people, so
![a=300,000](https://tex.z-dn.net/?f=a%3D300%2C000)
. Now, to convert the grow rate to decimal form, we are going to divide the rate by 100%:
![b= \frac{6.8}{100}](https://tex.z-dn.net/?f=b%3D%20%5Cfrac%7B6.8%7D%7B100%7D%20)
![b=0.068](https://tex.z-dn.net/?f=b%3D0.068)
Now that we have all the values we need, lets replace them in our grow function:
![y=a(1+b)^t](https://tex.z-dn.net/?f=y%3Da%281%2Bb%29%5Et)
![y=300,000(1+0.068)^t](https://tex.z-dn.net/?f=y%3D300%2C000%281%2B0.068%29%5Et)
![y=300,000(1.068)^t](https://tex.z-dn.net/?f=y%3D300%2C000%281.068%29%5Et)
We can conclude that the function that shows the relationship between
![y](https://tex.z-dn.net/?f=y)
and
![t](https://tex.z-dn.net/?f=t)
is
![y=300,000(1.068)^t](https://tex.z-dn.net/?f=y%3D300%2C000%281.068%29%5Et)
.