Question

the formula $a = 24.2e^{0.076t}$ 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 31.4 million?

a. in 2000, the population of the state was \\(\square\\) million.

Explanation:

Step1: Identify the value of t for 2000

Since the formula models the population \( t \) years after 2000, for the year 2000, \( t = 0 \).

Step2: Substitute t = 0 into the formula

The formula is \( A = 24.2e^{0.076t} \). Substituting \( t = 0 \), we get \( A = 24.2e^{0.076 \times 0} \).

Step3: Simplify the expression

We know that \( e^{0} = 1 \), so \( A = 24.2 \times 1 = 24.2 \).

Answer:

24.2