Question: The population of a certain inner-city area is estimated to be declining according to the model \(…

The population of a certain inner-city area is estimated to be declining according to the model \( P(t) = 390,000e^{-0.012t} \), where \( t \) is the number of years from the present. What does this model predict the population will be in 11 years? Round to the nearest person.

Answer

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people

Solution

To determine the population in 11 years using the given model, we use the formula: \[ P(t) = 390,000e^{-0.01t} \] Substitute \( t = 11 \) into the equation: \[ P(11) = 390,000e^{-0.01 \times 11} \] Calculate the exponent: \[ -0.01 \times 11 = -0.11 \] So, the equation becomes: \[ P(11) = 390,000e^{-0.11} \] Now, calculate \( e^{-0.11} \): \[ e^{-0.11} \approx 0.8958 \] Then multiply by 390,000: \[ P(11) = 390,000 \times 0.8958 \approx 349,362 \] The predicted population in 11 years, rounded to the nearest person, is 349,362 people.