Question

the population of a city with 15,000 people declines at a rate of 2% each year
Write the exponential equation that models the population, y after x years

Answer 1

To model the population decline of a city using an exponential equation, we can use the formula:

\[
y = y_0 \cdot (1 - r)^x
\]

Where:
- \(y\) is the population after \(x\) years,
- \(y_0\) is the initial population,
- \(r\) is the rate of decline (as a decimal),
- \(x\) is the number of years.

In this case:
- The initial population \(y_0 = 15000\),
- The rate of decline \(r = 0.02\) (since 2% is 0.02 as a decimal).

Now, substituting these values into the formula:

\[
y = 15000 \cdot (1 - 0.02)^x
\]

This simplifies to:

\[
y = 15000 \cdot (0.98)^x
\]

So the exponential equation that models the population \(y\) after \(x\) years is:

\[
y = 15000 \cdot (0.98)^x
\]