Expert: Josh - 7/23/2006

Question
i have read over the chapter for 2 hours and have no idea how to start this problem.

if you earned an average of 25000  over your working life and retire in 2005 at age 62,63, or 64 your annual social security would be $7000, $7500, or $8000. there is a linear equation that gives the annual benefit (b) in terms of age (a) for these 3 years. find the equation

Answer
Hi Sonya,

The average earning of 25000 is irrelevant as far as our modeling is concerned. It is just a condition which has to be satisfied, before someone is entitled to receive the social security benefit. What we should focus on, are the following figures:

(i) Age 62, 63 and 64 correspond to
(ii) 7000, 7500 and 8000, respectively.

If you like when age, a=62, b=7000; when a=63, b=7500; when a=64, b=8000.

We have been told that their relationship satisfies a linear equation, so, it will be of the form b=m*a+c, where m is the slope of the straight line; and "c" represents some constant.

What this means is that if we plug in, for instance, a=62, b=7000 into b=m*a+c, (7000=m*62+c) they should satisfy the equation.

Our task is to work out the slope "m" and b-intercept "c" (the point where the straight line crosses the vertical b-axis, when plotted against an horizontal a-axis.)

Remember this -- given two points (a1,b1) and (a2,b2), the slope may be computed as m=(b2-b1)/(a2-a1). Say, we pick (a1,b1) to be (62,7000) and pick (a2,b2) to be (64,8000).

The slope is very simply, m=(b2-b1)/(a2-a1)=(8000-7000)/(64-62)=1000/2=500. Since (a1,b1) must satisfy the equation b=m*a+c, substituting a1=62, b1=7000 gives 7000=500*62+c.

We solve for the unknown "c":
7000=31000+c,
c=7000-31000=-24000

Therefore, the straight line equation relating the age (a) and benefit (b) is given by:
b=500*a-24000

which may be factorized as b=500(a-48).

Hope this helps:)