Question

suppose you track the u.s. population over the past century and a half and find out that the function ( y = 0.0036e^{0.0126x} ), where x is the year, models it. the growth/decay factor for this function is (round to 4 decimal places). therefore, the population changes by % per year.

Explanation:

Step1: Identify growth factor form

Exponential growth function: $y = ab^{x} = ae^{kx}$, so $b = e^{k}$.

Step2: Calculate growth factor

$k = 0.0126$, so $b = e^{0.0126}$.

Step3: Compute value of $e^{0.0126}$

$e^{0.0126} \approx 1.01268$ (rounded to 5 decimals).

Step4: Round to 4 decimals

Growth factor $\approx 1.0127$.

Step5: Find percentage change

Percentage change = $(b - 1) \times 100\%$.

Step6: Calculate percentage

$(1.0127 - 1) \times 100\% = 1.27\%$.

Answer:

1.0127
1.27