Question

a town has a population of 72,700 and grows at a rate of 3.5% every year. which equation represents the town’s population after 4 years?
answer
\\( p = 72,700(1 - 0.035)^4 \\)
\\( p = 72,700(1 + 0.035)(1 + 0.035)(1 + 0.035) \\)
\\( p = 72,700(0.965)^4 \\)
\\( p = 72,700(1 + 0.035)(1 + 0.035)(1 + 0.035)(1 + 0.035) \\)

Explanation:

Step1: Identify growth formula logic

Population growth uses the compound growth principle, where each year's population is multiplied by $(1 + \text{growth rate})$. The growth rate is 3.5% = 0.035, so the multiplier per year is $1 + 0.035$.

Step2: Match to 4-year growth

For 4 years, the initial population ($72,700$) is multiplied by the annual growth factor 4 times. This can be written as repeated multiplication: $72,700(1 + 0.035)(1 + 0.035)(1 + 0.035)(1 + 0.035)$, or in exponent form $72,700(1 + 0.035)^4$ (the repeated multiplication form is one of the options).

Step3: Eliminate incorrect options

• Options with $(1 - 0.035)$ or $(0.965)$ represent population decline, which is not the case here.
• The option with only 3 multiplications of $(1 + 0.035)$ represents growth over 3 years, not 4.

Answer:

$P = 72,700(1 + 0.035)(1 + 0.035)(1 + 0.035)(1 + 0.035)$