12 mark for review the population on an island is modeled by p(t) = 6000 / (40 + 60e^(-0.1t)) for t ≥ 0…

12 mark for review the population on an island is modeled by p(t) = 6000 / (40 + 60e^(-0.1t)) for t ≥ 0, where p(t) is the number of people on the island after t years. what is lim p(t) as t→∞? a 60 b 100 c 150 d 6000

12 mark for review the population on an island is modeled by p(t) = 6000 / (40 + 60e^(-0.1t)) for t ≥ 0, where p(t) is the number of people on the island after t years. what is lim p(t) as t→∞? a 60 b 100 c 150 d 6000

Answer

Explanation:

Step1: Analyze the exponential - term as $t\to\infty$

As $t\to\infty$, the term $e^{- 0.1t}=\frac{1}{e^{0.1t}}\to0$ since the exponential function $y = e^{ax}$ with $a>0$ grows without bound as $x\to\infty$.

Step2: Evaluate the limit of $P(t)$

We have $P(t)=\frac{6000}{40 + 60e^{-0.1t}}$. Substituting the limit of $e^{-0.1t}$ as $t\to\infty$ into the formula for $P(t)$, we get $\lim_{t\to\infty}P(t)=\frac{6000}{40+60\times0}$.

Step3: Simplify the expression

$\frac{6000}{40 + 0}=\frac{6000}{40}=150$.

Answer:

C. 150