The number of U.S. citizens 65 and older from 1990 through 2050 is estimated to be growing at the rate of, R(t)=0.063t^2 -0.48t+3.87, (0 less than or = t less than or = 15) million people/decade, where t is measured in decades and t = 0 corresponds to 1900. Show that the average rate of growth of U.S. citizens 65 and older between 2000 and 2050 will be growing at almost twice the rate of that between 1950 and 2000.

Question

The number of U.S. citizens 65 and older from 1990 through 2050 is estimated to be growing at the rate of, R(t)=0.063t^2 -0.48t+3.87, (0 less than or = t less than or = 15) million people/decade, where t is measured in decades and t = 0 corresponds to 1900. Show that the average rate of growth of U.S. citizens 65 and older between 2000 and 2050 will be growing at almost twice the rate of that between 1950 and 2000.

Answer 1

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.

Answer 2

how did you get the 150? I was thinking it would be 50

Answer 3

since it's measured in decades

Answer 4

so it would not be 50 either but 5?

Answer 5

Oops. You are correct. So, make the fix and rerun the math.

Answer 6

To find the average rate of growth of U.S. citizens aged 65 and older between two periods, we need to calculate the rate of change of the population over those periods and compare them.

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.