Math Snap

STEP 1

1. We are given the formula $A = 19.6 e^{0.0139 t}$ which models the population $A$ in millions, $t$ years after 2000.
2. We need to find the population in the year 2000.
3. We need to determine when the population will reach 25.9 million.

STEP 2

1. Calculate the population in the year 2000.
2. Set up an equation to find when the population will reach 25.9 million.
3. Solve the equation for $t$.

STEP 3

To find the population in 2000, substitute $t = 0$ into the formula:
$$A = 19.6 e^{0.0139 \times 0}$$

STEP 4

Simplify the expression:
$$A = 19.6 e^0$$ $$A = 19.6 \times 1$$ $$A = 19.6$$ The population in the year 2000 was 19.6 million.

STEP 5

Set up the equation for when the population reaches 25.9 million:
$$25.9 = 19.6 e^{0.0139 t}$$

STEP 6

Solve for $t$. First, divide both sides by 19.6:
$$\frac{25.9}{19.6} = e^{0.0139 t}$$

STEP 7

Take the natural logarithm of both sides to solve for $t$:
$$\ln\left(\frac{25.9}{19.6}\right) = 0.0139 t$$

Solve for $t$ by dividing both sides by 0.0139:
$$t = \frac{\ln\left(\frac{25.9}{19.6}\right)}{0.0139}$$ The population of the state in 2000 was $\boxed{19.6}$ million.
To find when the population will reach 25.9 million, calculate $t$ using the equation from STEP_6.