The average rate of change between years a and b is
(R(b)-R(a))/(b-a)
Here, both intervals are 50 years, so things are convenient. You want to show that
R(150)-R(100) ≈ 2(R(100)-R(50))
So, just plug in your numbers and see what you get.
(R(b)-R(a))/(b-a)
Here, both intervals are 50 years, so things are convenient. You want to show that
R(150)-R(100) ≈ 2(R(100)-R(50))
So, just plug in your numbers and see what you get.
First, let's find the average rate of growth between 1950 and 2000. We are given the function R(t) = 0.063t^2 - 0.48t + 3.87, where t is measured in decades. We need to evaluate this function for the endpoints of the period.
For 1950 (t = 5), we have:
R(5) = 0.063(5)^2 - 0.48(5) + 3.87
= 0.063(25) - 2.4 + 3.87
= 1.575 - 2.4 + 3.87
= 2.045 million people/decade
For 2000 (t = 10), we have:
R(10) = 0.063(10)^2 - 0.48(10) + 3.87
= 0.063(100) - 4.8 + 3.87
= 6.3 - 4.8 + 3.87
= 5.37 million people/decade
The rate of growth between 1950 and 2000 is given by the difference in population divided by the difference in time:
Rate1 = (Population2 - Population1) / (Time2 - Time1)
= (5.37 - 2.045) / (10 - 5)
= 3.325 / 5
= 0.665 million people/decade
Now let's find the average rate of growth between 2000 and 2050. Again, we need to evaluate the function R(t) for the endpoints of the period.
For 2000 (t = 10), we already have the result: R(10) = 5.37 million people/decade.
For 2050 (t = 15), we can calculate:
R(15) = 0.063(15)^2 - 0.48(15) + 3.87
= 0.063(225) - 7.2 + 3.87
= 14.175 - 7.2 + 3.87
= 10.845 million people/decade
The rate of growth between 2000 and 2050 is given by:
Rate2 = (Population2 - Population1) / (Time2 - Time1)
= (10.845 - 5.37) / (15 - 10)
= 5.475 / 5
= 1.095 million people/decade
Now compare the two rates of growth, Rate1 and Rate2:
Rate1 / Rate2 = 0.665 / 1.095
≈ 0.607
The result shows that the average rate of growth of U.S. citizens aged 65 and older between 2000 and 2050 will be growing at approximately 0.607 times the rate of that between 1950 and 2000. In other words, the rate of growth between 2000 and 2050 is nearly twice the rate of growth between 1950 and 2000.