Question

the formula ( a = 24.2e^{0.0797t} ) models the population of a us state, ( a ), in millions, ( t ) years after 2000.

a. what was the population of the state in 2000?

b. when will the population of the state reach 31.4 million?

a. in 2000, the population of the state was 24.2 million.

b. the population of the state will reach 31.4 million in the year (square) (round to the nearest year as needed.)

Explanation:

Step1: Identify the formula and variables

The formula is \( A = 24.2e^{0.0797t} \), where \( A \) is the population in millions, and \( t \) is the number of years after 2000. For part b, we need to find \( t \) when \( A = 31.4 \).

Step2: Substitute \( A = 31.4 \) into the formula

We get the equation \( 31.4 = 24.2e^{0.0797t} \).

Step3: Divide both sides by 24.2

\( \frac{31.4}{24.2} = e^{0.0797t} \). Calculating the left side: \( \frac{31.4}{24.2} \approx 1.2975 \).

Step4: Take the natural logarithm of both sides

\( \ln(1.2975) = \ln(e^{0.0797t}) \). Using the property \( \ln(e^x) = x \), we have \( \ln(1.2975) = 0.0797t \).

Step5: Solve for \( t \)

\( t = \frac{\ln(1.2975)}{0.0797} \). Calculating \( \ln(1.2975) \approx 0.260 \), then \( t \approx \frac{0.260}{0.0797} \approx 3.26 \).

Step6: Find the year

Since \( t \) is years after 2000, the year is \( 2000 + 3.26 \approx 2003 \) (rounded to the nearest year).

Answer:

2003