Question

the population of a certain inner-city area is estimated to be declining according to the model ( p(t) = 580,000e^{-0.022t} ), where ( t ) is the number of years from the present. what does this model predict the population will be in 5 years? round to the nearest person.

Explanation:

Step1: Identify the value of t

We need to find the population in 5 years, so \( t = 5 \).

Step2: Substitute t into the population model

The population model is \( P(t)=580000e^{- 0.022t} \). Substitute \( t = 5 \) into the model:
\( P(5)=580000e^{-0.022\times5} \)

Step3: Calculate the exponent

First, calculate the value of the exponent: \( - 0.022\times5=-0.11 \)
So the formula becomes \( P(5)=580000e^{-0.11} \)

Step4: Calculate the value of \( e^{-0.11} \)

We know that \( e^{-0.11}\approx0.8958 \) (using a calculator to find the value of the exponential function)

Step5: Calculate the population

Multiply 580000 by 0.8958:
\( P(5)=580000\times0.8958 = 580000\times0.8958=519564 \)

Answer:

519564