Question

The population of a certain city grows exponentially. When the city was founded, it only had 1000 residents. After 1 year the city had 1050 residents. Write an equation that models the population y of the city after x years.

Answer 1

Our function is something like

`y=ae^(bx)`

When the city was founded, 0 years have passed. So, we want

`y(0)=1000 \iff ae^(0x)=1000 \iff a\cdot 1 = 1000 \iff a=1000`

So, we can update our function: now we know that it will be something like

`y=1000e^(bx)`

Similarly, we have to impose
`y(1)=1050`. Since

`y(1)=1000e^(b\cdot 1)=1000e^b`

we have to impose

`1000e^b=1050 \iff e^b = 1.05 \iff b = \ln(1.05)`

So, our function is

`y=1000e^(\ln(1.05)x)`