2. the population of sacramento increases by 2% each year. right now, the population is about 500,930. what will the population of sacramento be in 2030? 100% + 2 = 102%. y = a. b ^ { x } y = 500,930 ( 1.02 ) ^ { 5 } = 553067.1967

Question

Accepted Answer

# Explanation:
## Step1: Determine the growth factor
The population increases by $2\%$ each year. The growth factor $b = 1+ 0.02=1.02$. The initial population $a = 500930$.
## Step2: Determine the number of years $x$
Assume the current year is $2025$. The number of years from $2025$ to $2030$ is $x = 2030 - 2025=5$
## Step3: Use the exponential growth formula $y=a\times b^{x}$
Substitute $a = 500930$, $b = 1.02$ and $x = 5$ into the formula $y=500930\times(1.02)^{5}$
$$
\begin{align*}
(1.02)^{5}&=(1 + 0.02)^{5}\
&=\sum_{k = 0}^{5}\binom{5}{k}(0.02)^{k}(1)^{5 - k}\
&=1+5\times0.02 + 10\times(0.02)^{2}+10\times(0.02)^{3}+5\times(0.02)^{4}+(0.02)^{5}\
&=1 + 0.1+0.004 + 0.00008+0.0000016+0.000000032\
&=1.104081632
\end{align*}
$$
$y=500930\times1.104081632\approx553067$

# Answer:
The population of Sacramento in $2030$ will be approximately $553067$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Determine the growth factor

Step2: Determine the number of years \(x\)

Step3: Use the exponential growth formula \(y=a\times b^{x}\)

Answer: