Question

the llama population on an island is decreasing by 10 percent per year. use an exponential equation find the population after 10 years if the population started with 50,000 llamas at the end of year 1. (1 point) 19,371 49,900 15,690 17,433 check answer remaining attempts : 3

Explanation:

Step1: Recall exponential decay formula

The formula for exponential decay is $P(t) = P_0(1 - r)^t$, where $P_0$ is the initial population, $r$ is the rate of decay (as a decimal), and $t$ is the time in years.
Here, $P_0 = 50000$, $r = 0.10$ (since 10% = 0.10), and we need to find the population after $t = 10 - 1 = 9$ years? Wait, no—wait, the problem says "at the end of year 1" the population is 50,000. So we need to find the population after 10 years, which is 9 years after year 1. Wait, no, let's re - read: "the population started with 50,000 llamas at the end of year 1. Use an exponential equation find the population after 10 years". So the time elapsed from year 1 to year 10 is $t=10 - 1=9$ years? Wait, no, maybe I misread. Wait, if at the end of year 1, the population is 50,000, and we want the population after 10 years (from the start? No, the problem says "after 10 years" given that at the end of year 1 it's 50,000. So the number of years of decay is $10 - 1=9$? Wait, no, maybe the initial time is year 1, and we want year 10, so $t = 9$? Wait, no, let's check the formula again. The general formula for exponential decay is $P(t)=P_0(1 - r)^t$, where $t$ is the number of time periods. If at $t = 0$ (start), but here at the end of year 1 ($t = 1$) the population is 50,000. So we can model it as $P(t)=50000(1 - 0.10)^{t - 1}$, where $t$ is the year number. We want $t = 10$, so $P(10)=50000(0.90)^{10 - 1}=50000(0.90)^9$.

Step2: Calculate $(0.90)^9$

First, calculate $0.9^9$. We know that $0.9^1 = 0.9$, $0.9^2=0.81$, $0.9^3 = 0.729$, $0.9^4=0.6561$, $0.9^5 = 0.59049$, $0.9^6=0.531441$, $0.9^7 = 0.4782969$, $0.9^8=0.43046721$, $0.9^9 = 0.387420489$.

Step3: Calculate $P(10)$

Now, multiply by 50,000: $P(10)=50000\times0.387420489 = 19371.02445\approx19371$.

Answer:

19,371