Question

the formula a = 15.7e^{0.0405t} 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 18.7 million?

Explanation:

Step1: Find population in 2000

In 2000, \(t = 0\). Substitute \(t = 0\) into \(A=15.7e^{0.0405t}\).
\[A = 15.7e^{0.0405\times0}=15.7e^{0}\]
Since \(e^{0}=1\), then \(A = 15.7\times1=15.7\).

Step2: Find \(t\) when \(A = 18.7\)

Set \(A = 18.7\) in the formula \(A = 15.7e^{0.0405t}\), we get \(18.7=15.7e^{0.0405t}\).
First, divide both sides by 15.7: \(\frac{18.7}{15.7}=e^{0.0405t}\), so \(e^{0.0405t}=\frac{18.7}{15.7}\approx1.1911\).
Then take the natural - logarithm of both sides: \(\ln(e^{0.0405t})=\ln(1.1911)\).
Using the property \(\ln(e^{x}) = x\), we have \(0.0405t=\ln(1.1911)\).
Since \(\ln(1.1911)\approx0.175\), then \(t=\frac{\ln(1.1911)}{0.0405}=\frac{0.175}{0.0405}\approx4.32\).

Answer:

a. 15.7 million
b. Approximately 4.32 years after 2000 (or around 2004)