Question

a town has a population of $2.35 \times 10^4$ and shrinks at a rate of 9.1% every year. which equation represents the town’s population after 2 years?
answer
\\( \circ \\ p = (2.35 \times 10^4)(1 + 0.091)^2 \\)
\\( \circ \\ p = (2.35 \times 10^4)(0.09)^2 \\)
\\( \circ \\ p = (2.35 \times 10^4)(1 - 0.091)(1 - 0.091)(1 - 0.091)(1 - 0.091) \\)
\\( \circ \\ p = (2.35 \times 10^4)(0.909)^2 \\)

Explanation:

Step1: Identify decay rate factor

The population shrinks by 9.1%, so the remaining population each year is $1 - 0.091 = 0.909$.

Step2: Set up decay formula

For exponential decay over $t$ years, the formula is $P = P_0(1 - r)^t$, where $P_0=2.35\times10^4$, $r=0.091$, $t=2$. Substitute values:
$P = (2.35\times10^4)(0.909)^2$

Step3: Eliminate incorrect options

• Option1 uses growth factor $(1+0.091)$, wrong for shrinkage.
• Option2 uses only the decay rate, not remaining population.
• Option3 applies decay 4 times, not 2 years.

Answer:

$P = (2.35 \times 10^4)(0.909)^2$