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?
a. in 2000, the population of the state was 15.7 million.
b. the population of the state will reach 18.7 million in the year (round to the nearest year as needed.)

Explanation:

Step1: Set A = 18.7 in the formula

$18.7 = 15.7e^{0.0405t}$

Step2: Isolate the exponential term

$\frac{18.7}{15.7}=e^{0.0405t}$
$1.191083 = e^{0.0405t}$

Step3: Take the natural - logarithm of both sides

$\ln(1.191083)=\ln(e^{0.0405t})$
Since $\ln(e^{x}) = x$, we have $\ln(1.191083)=0.0405t$

Step4: Solve for t

$t=\frac{\ln(1.191083)}{0.0405}$
$\ln(1.191083)\approx0.175$
$t=\frac{0.175}{0.0405}\approx4.32$

Step5: Find the year

The year is $2000 + 4.32\approx2004$

Answer:

2004