Question

the formula ( a = 18.5e^{0.1704t} ) 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
the population of the state will reach 26.6 million in the year (square).
(round up to the nearest year.)

Explanation:

Step1: Substitute \(A = 26.6\) into the formula

Given \(A=18.5e^{0.1704t}\), substitute \(A = 26.6\):
\(26.6=18.5e^{0.1704t}\)

Step2: Solve for \(e^{0.1704t}\)

Divide both sides by \(18.5\):
\(\frac{26.6}{18.5}=e^{0.1704t}\)
\(1.437838 = e^{0.1704t}\)

Step3: Take the natural logarithm of both sides

Since \(y = e^{x}\) implies \(\ln(y)=x\), we have:
\(\ln(1.437838)=0.1704t\)
We know that \(\ln(1.437838)\approx0.363\)

Step4: Solve for \(t\)

\(t=\frac{\ln(1.437838)}{0.1704}\)
Substitute \(\ln(1.437838)\approx0.363\)
\(t=\frac{0.363}{0.1704}\approx 2.13\)

Answer:

The population of the state will reach \(26.6\) million in the year \(2002\) (since \(t\approx2.13\) and we round up to the nearest year, and \(2000 + 2=2002\))