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

$y=1000e^{\ln(1.05)x}$