Question

the fish population in a lake can be modeled by the function $p(x) = 3,640(1.15)^x$, where $x$ represents the number of years since 2000. which statement best interprets one value in the function? a the fish population increases at a rate of 36.40% per year. b the initial fish population is 4,185. c the initial fish population is 115. d the fish population increases at a rate of 15% per year.

Explanation:

Step1: Recall exponential growth form

The standard exponential growth function is $P(x) = P_0(1+r)^x$, where $P_0$ is the initial population, $r$ is the annual growth rate, and $x$ is time in years.

Step2: Match given function to standard form

Given $P(x) = 3,640(1.15)^x$, compare to $P_0(1+r)^x$:

• $P_0 = 3640$ (initial population)
• $1+r = 1.15$, so $r = 1.15 - 1 = 0.15$ or 15%

Step3: Evaluate each option

• Option A: Incorrect, growth rate is 15%, not 36.40%.
• Option B: Incorrect, initial population is 3640, not 4186.
• Option C: Incorrect, initial population is 3640, not 115.
• Option D: Correct, $r=0.15$ means 15% annual growth rate.

Answer:

D. The fish population increases at a rate of 15% per year.