Question

The formula Upper A equals 18.5 e Superscript 0.1724 t models the population of a US​ state, A, in​ millions, t years after 2000. Determine algebraically when the population was predicted to reach 26.6 million.
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Part 1
The population of the state will reach 26.6 million in the year
  
enter your response here.
​(Round up to the nearest​ year.)

Answer 1

To determine when the population \( A \) will reach 26.6 million, we start with the equation given in the problem:

\[
A = 18.5 e^{0.1724t}
\]

We need to find \( t \) when \( A = 26.6 \) million. Thus, we set up the equation:

\[
26.6 = 18.5 e^{0.1724t}
\]

Now, we will isolate \( e^{0.1724t} \):

1. Divide both sides by 18.5:

\[
\frac{26.6}{18.5} = e^{0.1724t}
\]

2. Calculate \( \frac{26.6}{18.5} \):

\[
\frac{26.6}{18.5} \approx 1.44
\]

Now, we have:

\[
1.44 = e^{0.1724t}
\]

3. Take the natural logarithm (ln) of both sides to solve for \( t \):

\[
\ln(1.44) = 0.1724t
\]

4. Now, solve for \( t \):

\[
t = \frac{\ln(1.44)}{0.1724}
\]

5. Calculate \( \ln(1.44) \):

\[
\ln(1.44) \approx 0.362
\]

6. Substitute \( \ln(1.44) \) into the equation for \( t \):

\[
t = \frac{0.362}{0.1724} \approx 2.10
\]

Therefore, rounding up to the nearest year, \( t \) is approximately 3.

To find the corresponding year, we add \( t \) to 2000:

\[
2000 + 3 = 2003
\]

Thus, the population of the state will reach 26.6 million in the year **2003**.