Question

the population of a town is decreasing at a rate of 1% per year in 2000 there were 1300 people write an exponential decay function to model this situation

Answer 1

To model the population decay of the town, we can use the exponential decay function given by:

\[ P(t) = P_0 e^{kt} \]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.

Since the population is decreasing at a rate of 1% per year, we can express the decay constant \( k \) as:

\[ k = -0.01 \]

This negative sign indicates a decrease in population.

Given that the initial population \( P_0 \) in the year 2000 was 1300, we can substitute these values into the function:

\[ P(t) = 1300 e^{-0.01t} \]

Where \( t \) is the number of years since 2000.

So, the exponential decay function that models the population of the town is:

\[ P(t) = 1300 e^{-0.01t} \]

This function can be used to calculate the population at any given year \( t \). For example, for the year 2020 (which is 20 years after 2000), you would substitute \( t = 20 \) into the equation to find the population at that time.