1. In a certain country, the percentage of the population relocating to a new town is given by the following function:
p(t) = 20.6e^-0,009t,where 0 ¡Ü t ¡Ü 52.
Time is zero (t = 0) corresponds to the year 1960.
1. Find the rate at which people relocated during the year of your birth.
2. Find the relocation rate of today.
What does this tell you about the country¡¯s population? Is there a peak (max) relocation year? Do you think this model is appropriate for predicting population movement? Share and compare calculations with another classmate
To find the rate at which people relocated during the year of your birth, you need to substitute the value of t that corresponds to the year of your birth into the given function p(t). Since time is measured in years and t = 0 corresponds to the year 1960, you need to subtract your birth year from 1960 to find the value of t.
For example, let's say you were born in 1990. Subtracting 1990 from 1960 gives us t = -30. Now, substitute this value into the function p(t):
p(-30) = 20.6e^(-0.009 * -30)
Next, evaluate the exponential term:
p(-30) = 20.6e^(0.27)
Using a calculator, find the value of e^0.27 (around 1.309). Multiply this value by 20.6 to get the relocation rate during the year of your birth. So the rate at which people relocated during the year of your birth is approximately 26.95%.
Now, to find the relocation rate today, substitute the current year into the function p(t) following the same procedure as above. Let's say the current year is 2022. Subtracting 2022 from 1960 gives us t = 62. Substitute this value into the function p(t):
p(62) = 20.6e^(-0.009 * 62)
Evaluate the exponential term:
p(62) = 20.6e^(-0.558)
Using a calculator, find the value of e^-0.558 (around 0.572). Multiply this value by 20.6 to get the current relocation rate. So the relocation rate today is approximately 11.81%.
Based on these calculations, we can see that the relocation rate has decreased over time. This suggests that the country's population is becoming more settled and less likely to relocate.
To determine if there is a peak (max) relocation year, you can analyze the trend of the function p(t). By taking the derivative of p(t) with respect to t and finding its critical points, you can identify the year(s) with the maximum relocation rate. However, since the exact function p(t) is not provided, it is not possible to find the peak relocation year without additional information.
Regarding whether this model is appropriate for predicting population movement, it largely depends on the context and the data being used. The given function assumes an exponential decay of relocation rate over time. This may be suitable for certain scenarios where population movement tends to decrease over time, such as when urbanization or economic stability increases. However, it may not accurately represent situations where factors like migration patterns, economic fluctuations, or policy changes significantly influence population movement. Therefore, it is important to consider the limitations and assumptions of the model when using it for predictions and to compare results with real-world data and other models to assess its accuracy.