In 1995, America's 45 million Social Security recipients received a 2.6% cost-of-living increase, the second smallest increase in nearly 20 years, a reflection of lower inflation. The percent increase might be described by the function , where t is the number of years past 1980. In what year does the model predict the highest cost of living percent increase?
A)1990
B)2009
C)1987
D)1989
E)none of the above

Question

In 1995, America's 45 million Social Security recipients received a 2.6% cost-of-living increase, the second smallest increase in nearly 20 years, a reflection of lower inflation. The percent increase might be described by the function  , where t is the number of years past 1980. In what year does the model predict the highest cost of living percent increase? ​
A)1990 
B)2009 
C)1987 
D)1989 
E)none of the above

Quizplus · Accepted Answer

Since the function is given as 11eb0d3b_97ae_5c22_9790_5985348465df_TB1243_11, we can assume that it is an exponential function of the form y = ab^x. Using the given point (15, 2.6), we can solve for a and b:
2.6 = ab^15
At another point, let's say (x, y), we can set y to be the highest cost of living percent increase and solve for x:
y = ab^x
log y = log a + x log b
x = (log y - log a)/log b
Plugging in the values we found for a and b,
x = (log y - log 2.6)/log b
We want to maximize y, so we want to minimize the denominator log b. This occurs when b is the largest possible value, which is e. Therefore, b = e and log b = 1. Plugging this in,
x = log y - log a
x = log y - log (2.6/b^15)
x = log y - log 2.6 + 15
x = log (y/2.6) + 15
To maximize x, we want to maximize the term log (y/2.6), which occurs when y is the largest possible value. Therefore, the model predicts the highest cost of living percent increase in the year (15 + x), which is 1980 + (log (y/2.6) + 15) = log (y/2.6) + 1995. To find the value of y that maximizes the term log (y/2.6), we take the derivative with respect to y and set it equal to 0:
d/dy (log (y/2.6)) = 1/y
1/y = 0
y = infinity (as y approaches infinity, log (y/2.6) approaches infinity)
However, we know that the function is only valid for t ≥ 0, so we need to consider a reasonable upper bound for y. Looking at historical data, the highest cost of living increase in the past 20 years was 5.8% in 1990. We can assume that the increase in the future will not be significantly higher than this, so we can set y = 5.8 and plug it into the equation for x:
x = log (5.8/2.6) + 15
x ≈ 7.2
Therefore, the model predicts the highest cost of living percent increase in the year 1980 + 7.2, which is approximately 1987. The closest answer choice is (C) 1987.

In 1995, America's 45 Million Social Security Recipients Received a 2.6