Question

1. Identify what they are given and what they need to find;

2. Identify the type of problem they have been given and the tools necessary to solve the problem;

3. Correctly apply the tools to the information given to set up the problem;

4. Perform mathematically correct calculations to determine a solution;

5. Interpret their results in terms of the original problem.

Use the internet and find the Medicare expenditures in the year 2000 and the Medicare expenditures in 2015. Use the exponential growth function and develop a model. Let t = 0 in the year 2000. If the model remains accurate, estimate Medicare expenditures for the year 2017.

Answer 1

The Medicare expenditures in USA in the years 2000 and 2015 were $ 206.2 billion and $ 552.3 billion respectively which indicate an exponential growth. Let the exponential growth model for the Medicare expenditure M(t) (say) be M(t) = ab^t , where a,b are arbitrary real numbers and t is the number of years from 2000. In the year 2000, when t = 0, we have ab⁰ = 206.2 or, a = 206.2 ( as b⁰ =1). Now, on substituting a = 206.2, the growth model changes to M(t) = 206.2 b^t. Further, in the year 2015, when t = 15, we have 206.2 b¹⁵ = 552.3 so that b¹⁵ = 552.3/206.2. Then b = (552.3/206.2)^1/15 = 1.067888129 = 1.0679( on rounding off to 4 decimal places). Now, on substituting b =0.916, the growth model changes to M(t) = 206.2 (1.0679)^t.

Now, we will check this growth model for accuracy. As per this growth model, the Medicare expenditure in the year 2017 (when t = 17) would be 206.2*(1.0679)¹⁷ = $ 629.83 billion.

As per the information available on the internet, the actual Medicare expenditure in the year 2017 was $ 595.5 billion.

Thus, the exponential growth model worked out as above, is not accurate for predicting future growth. The possible reasons could be rounding off errors and also determining the exponential growth model on the basis of only 2 data points.