R (t) = 0.00731t^4 – 0.174t^3 + 1.528t^2 + 0.48t + 19.3; (0=t=10)

Question

The share of the world population that is over 60 years of age compared to the rest of the working population in the world is of concern to economists. An increasing dependency ration means that there will be fewer workers to support an aging population. The dependency ration over the next century is forecast to be:
R (t) = 0.00731t^4 – 0.174t^3 + 1.528t^2 + 0.48t + 19.3; (0≤t≤10)
In year t, where t is measured in decades with t = 0 corresponding to 2000.

a. Show that the dependency ration will be increasing at the fastest pace around 2052.
(Hint : Use the quadratic formula)

b. What will the dependency ration be at that time?

Answer 1

R(t) = 0.00731 t⁴ – 0.174 t³ + 1.528 t² + 0.48 t + 19.3

 

At what rate will the dependency ratio be increasing?  That's just the derivative of R(t):

 

R'(t) = 0.02924 t³ - 0.522 t² + 3.056 t + 0.48.

 

They want you to find the maximum of the rate of increase.  To find the maximum, you have to differentiate again:

 

R''(t) = 0.08772 t² -1.044 t + 3.056

 

...then set the derivative to 0 and solve for t:

 

0.08772 t² -1.044 t + 3.056 = 0

 

Using the quadratic formula, there are two solutions for t:  6.7079 and 5.1936

 

Now we have to find out if either of these is a maximum.  To do this you take the derivative of R''(t).  If it is negative at one of these values, then that point is a maximum.

 

So R'''(t) = 0.17544 t - 1.044

 

R'''(6.7079) = 0.132833976

R'''(5.1936) = -0.132834816

 

So t = 5.1936 is a maximum, and is the point we are looking for.  Since t is measured in decades, that represents 51.936 years after 2000, which is the year 2052, when you round to the nearest year.

 

 

To answer part b, just substitute 5.1936 back into the original formula for R(t).  I'll leave that to you, since it's just a matter of using a calculator.