Write an exponential function for the situation below and use your equation to answer the question.

Question

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The population of a town is 6,000 and is decreasing at a rate of 3% each year. What will the town's population be after five years?

Answer 1

To model the situation with an exponential function, we can use the formula for exponential decay:

\[ P(t) = P_0 \times (1 - r)^t \]

Where:

• \( P(t) \) is the population at time \( t \),
• \( P_0 \) is the initial population,
• \( r \) is the rate of decay (expressed as a decimal),
• \( t \) is the time in years.

Given:

• Initial population \( P_0 = 6000 \),
• Rate of decay \( r = 0.03 \) (3%),
• Time \( t = 5 \) years.

Plugging in the values, we get:

\[ P(5) = 6000 \times (1 - 0.03)^5 \]

Calculating it step by step:

1. Calculate \( 1 - 0.03 = 0.97 \).
2. Raise \( 0.97 \) to the power of \( 5 \):

\[ 0.97^5 \approx 0.8597 \]

3. Multiply by \( 6000 \):

\[ P(5) \approx 6000 \times 0.8597 \approx 5158.2 \]

Therefore, after five years, the population of the town will be approximately 5,158.

In summary, the exponential function representing the population decay scenario is:

\[ P(t) = 6000 \times (0.97)^t \]

And after 5 years, the population is approximately 5,158.