Question

the population of a certain inner - city area is estimated to be declining according to the model ( p(t)=466,000e^{-0.018t} ), where ( t ) is the number of years from the present. what does this model predict the population will be in 12 years? round to the nearest person.

Explanation:

Step1: Identify the formula and values

We have the population model \( P(t) = 466000e^{-0.018t} \), and we need to find \( P(12) \), so \( t = 12 \).

Step2: Substitute \( t = 12 \) into the formula

Substitute \( t = 12 \) into \( P(t) \): \( P(12)=466000e^{-0.018\times12} \)
First, calculate the exponent: \( - 0.018\times12=-0.216 \)
So, \( P(12) = 466000e^{-0.216} \)

Step3: Calculate the value of \( e^{-0.216} \)

We know that \( e^{-0.216}\approx e^{- 0.216}\approx0.8067 \) (using a calculator to find the value of the exponential function)

Step4: Calculate the population

Multiply \( 466000 \) by \( 0.8067 \): \( 466000\times0.8067 = 466000\times0.8067 = 375922.2\)

Step5: Round to the nearest person

Rounding \( 375922.2 \) to the nearest person gives \( 375922 \)

Answer:

\( 375922 \)