Question

after years of maintaining a steady population of 32, 000, the population of a town begins to grow exponentially. after 1 year and an increase of 8% per year, the population is 34, 560. which equation can be used to predict, y, the number of people living in the town after x years? (round population values to the nearest whole number.)
$y = 34,560(1.08)^x$
$y = 34,560(0.08)^x$
$y = 32,000(0.08)^x$

Explanation:

Step1: Recall exponential growth formula

The standard exponential growth formula is $y = a(1+r)^x$, where $a$ is the initial amount for the model, $r$ is the annual growth rate, and $x$ is time in years.

Step2: Identify correct initial value

Here, the starting population for the growth model is the population after 1 year, $a=34,560$, and $r=0.08$, so $1+r=1.08$.

Step3: Eliminate incorrect options

• $y=34,560(0.08)^x$ uses a decay factor, not growth.
• $y=32,000(0.08)^x$ uses the original population and a decay factor, which is wrong.

Answer:

$y = 34, 560(1.08)^x$