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