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)
Responses
17,433
17,433
15,690
15,690
19,371
19,371
49,900
Responses
17,433
17,433
15,690
15,690
19,371
19,371
49,900
Answers
GPT-4o mini
To find the llama population after 10 years with an initial population of 50,000 and a decrease of 10% per year, we can use the formula for exponential decay:
\[
P(t) = P_0 \times (1 - r)^t
\]
Where:
- \( P(t) \) = the population at time \( t \)
- \( P_0 \) = the initial population
- \( r \) = the rate of decrease (as a decimal)
- \( t \) = time in years
Given:
- \( P_0 = 50,000 \)
- \( r = 0.10 \)
- \( t = 10 \)
Now we can plug in the values:
\[
P(10) = 50,000 \times (1 - 0.10)^{10}
\]
Calculating:
\[
P(10) = 50,000 \times (0.90)^{10}
\]
Now compute \( (0.90)^{10} \):
\[
(0.90)^{10} \approx 0.3487
\]
Now calculate \( P(10) \):
\[
P(10) \approx 50,000 \times 0.3487 \approx 17,435
\]
The calculated population after 10 years, rounded to the nearest whole number, is approximately **17,435**. Among the provided options, the closest value is:
**17,433**
\[
P(t) = P_0 \times (1 - r)^t
\]
Where:
- \( P(t) \) = the population at time \( t \)
- \( P_0 \) = the initial population
- \( r \) = the rate of decrease (as a decimal)
- \( t \) = time in years
Given:
- \( P_0 = 50,000 \)
- \( r = 0.10 \)
- \( t = 10 \)
Now we can plug in the values:
\[
P(10) = 50,000 \times (1 - 0.10)^{10}
\]
Calculating:
\[
P(10) = 50,000 \times (0.90)^{10}
\]
Now compute \( (0.90)^{10} \):
\[
(0.90)^{10} \approx 0.3487
\]
Now calculate \( P(10) \):
\[
P(10) \approx 50,000 \times 0.3487 \approx 17,435
\]
The calculated population after 10 years, rounded to the nearest whole number, is approximately **17,435**. Among the provided options, the closest value is:
**17,433**