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
p ' (t) = (-.009)(20.6) e^(-.009t)
plug in your birth-year. Sorry, I don't know it, you didn't invite me to your birthday party.
"today" ---> t = 53 , which is beyond the domain given. (Time to get a new edition of the textbook)
if you want to do it anyway
p ' (53) = (-.009)(20.6) e^(-.009(53) )
= appr -.115
max relocation ----> set p ' (t) = 0
(-.009)(20.6) e^(-.009t) = 0
e^(-.009t) = 0
no solution, thus no max
p'(53) = (-.009)(20.6)e^(-.009(53))
= -.1854e^((-.477)
= -.1854 * .62064
= -.115067
= -.115067 * 100
= -11.5067 % change in
relocation rate
To find the rate at which people relocated during the year of your birth, you need to calculate the derivative of the given function with respect to time (t) and evaluate it at the specific year.
1. To calculate the derivative, use the power rule and the chain rule. The derivative of p(t) = 20.6e^-0.009t is given by:
p'(t) = 20.6 * (-0.009) * e^(-0.009t)
2. Substitute the specific year of your birth into the derivative equation and evaluate it to find the relocation rate during that year.
For example, if your birth year is 1990, the corresponding value of t would be 1990 - 1960 = 30.
p'(30) = 20.6 * (-0.009) * e^(-0.009 * 30)
Evaluate the expression to get the relocation rate during the year of your birth.
To find the relocation rate of today, substitute the current year into the derivative equation and evaluate it.
For example, if the current year is 2021, the corresponding value of t would be 2021 - 1960 = 61.
p'(61) = 20.6 * (-0.009) * e^(-0.009 * 61)
Evaluate the expression to get the relocation rate today.
By analyzing the relocation rates, you can make conclusions about the country's population. If the relocation rate during the year of your birth is high, it suggests that many people were moving to new towns then. If the relocation rate is high today, it indicates a significant movement of people currently.
To determine if there is a peak (max) relocation year, you can find the maximum value of the relocation function or its derivative. The year corresponding to this maximum value would be the peak relocation year if such exists.
Regarding the appropriateness of this model for predicting population movement, it depends on various factors. This exponential decay model assumes a certain behavior of the population relocation rate over time. However, reality may involve more complex dynamics that the model does not capture. Comparing calculations with another classmate can help validate and compare the results obtained using this model.