To model the population of a city with an initial population of 15,000 people that declines at a rate of 2% each year using an exponential equation, you can use the following formula:
\[ y(t) = y_0 \cdot (1 - r)^t \]
Where:
- \( y(t) \) is the population after \( t \) years,
- \( y_0 \) is the initial population (15,000 in this case),
- \( r \) is the rate of decline (expressed as a decimal),
- \( t \) is the number of years.
Given that the decline rate is 2%, we have:
- \( y_0 = 15000 \)
- \( r = 0.02 \)
Substituting these values into the equation gives:
\[ y(t) = 15000 \cdot (1 - 0.02)^t \]
This simplifies to:
\[ y(t) = 15000 \cdot (0.98)^t \]
So, the exponential equation that models the population \( y \) after \( t \) years is:
\[ y(t) = 15000 \cdot (0.98)^t \]
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