the formula a = 15.7e^{0.0405t} models the population of a us state, a, in millions, t years after 2000.\na…

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)