Problem 70 - College Algebra Essentials

Question

Although Social Security is a problem, some projections indicate that there's a much bigger time bomb ticking in the federal budget, and that's Medicare. In 2000 , the cost of Social Security was \(5.48 \%\) of the gross domestic product, increasing by \(0.04 \%\) of the GDP per year. In \(2000,\) the cost of Medicare was \(1.84 \%\) of the gross domestic product, increasing by \(0.17 \%\) of the GDP per year. a. Write a function that models the cost of Social Security as a percentage of the GDP \(x\) years after 2000 . b. Write a function that models the cost of Medicare as a percentage of the GDP \(x\) years after 2000 . c. In which year will the cost of Medicare and Social Security be the same? For that year, what will be the cost of each program as a percentage of the GDP? Which program will have the greater cost after that year?

Step-by-Step Solution

Answer

In 2028, both Social Security and Medicare are projected to cost 6.60% of the GDP. From then on, Medicare will have a greater cost than Social Security.

Step 1: Formulate cost functions

Given that the cost of Social Security was 5.48% of GDP in 2000 and it increases by 0.04% per year, the function modeling the cost of Social Security \(S\) as a percentage of GDP \(x\) years after 2000 can be written as,\(S(x) = 5.48 + 0.04x\).Similarly, the cost of Medicare was 1.84% of the GDP in 2000 and increases by 0.17% each year. Therefore, the function \(M(x)\) modeling the cost of Medicare is,\(M(x) = 1.84 + 0.17x\).

Step 2: Solve for Equal Costs

We determine the year when the Social Security and Medicare costs will be the same by setting the two functions equal to each other and solving for \(x\):\(5.48 + 0.04x = 1.84 + 0.17x\). We subtract 0.04x from both sides and subtract 1.84 from both sides to isolate \(x\) and obtain,\(0.13x = 3.64\). Dividing both sides by 0.13 results in,\(x = 28\). So, 28 years after 2000, which is 2028, the costs will be the same.

Step 3: Determine Costs and Greater Costing Program

To find the costs of each program as a percentage of the GDP in that year, we substitute \(x = 28\) into both functions,Thus, \(S(28) = 5.48 + 0.04*28 = 6.60\%\), and\(M(28) = 1.84 + 0.17*28 = 6.60\%\).Both costs will be equal to \(6.60\%\) of GDP in 2028. After 2028, the Medicare program will have a greater cost because its rate of cost increase is higher than that of Social Security.

Step 4: Conclusion

In conclusion, the costs of Social Security and Medicare will align in 2028 with each accounting for \(6.60\%\) of the GDP. After this point, Medicare will exceed Social Security in terms of cost to GDP ratio.

Key Concepts

Modeling growth

Modeling growth is essential in understanding how particular variables evolve over time. In our context, the growth model helps to project the annual increase in costs as a percentage of GDP for Social Security and Medicare. The function for Social Security is modeled as follows:

• Initial cost in 2000: 5.48% of GDP
• Increase: 0.04% per year

This gives the growth model:\[ S(x) = 5.48 + 0.04x \]where \( S(x) \) represents the cost of Social Security \( x \) years after 2000. Similarly, the function for Medicare is:

• Initial cost in 2000: 1.84% of GDP
• Increase: 0.17% per year

This results in the function:\[ M(x) = 1.84 + 0.17x \]where \( M(x) \) indicates the cost of Medicare \( x \) years after 2000. By understanding these models, we can assess future economic commitments.

Solving linear equations

Solving linear equations is a critical skill in algebra used to find unknown variables by forming equalities. Here, we use this skill to find when Social Security and Medicare costs will be the same. We express both costs with functions and set them equal:\[ 5.48 + 0.04x = 1.84 + 0.17x \]To solve this, we simplify by eliminating \( x \) terms from one side and numbers from the other:- Subtract 0.04x from both sides: \( 5.48 = 1.84 + 0.13x \)- Subtract 1.84 from both sides: \( 3.64 = 0.13x \)Finally, divide both sides by 0.13 to isolate \( x \):\[ x = 28 \]Thus, after 28 years from 2000, or in 2028, the cost will be the same. This is a simple way of finding when two different circumstances line up.

Functions equality

Understanding the concept of functions equality allows us to determine when two different functions give the same output. With our cost functions \( S(x) \) and \( M(x) \), we found they equalize in 2028:

• Social Security: \( S(28) = 6.60\% \)
• Medicare: \( M(28) = 6.60\% \)

When two functions intersect, it indicates a notable change or equilibrium like in our scenario. Both Social Security and Medicare costs will be the same percentage of GDP in that particular year. After that year, understanding the differing rates of increase between the two can predict which will become more costly.

GDP cost analysis

GDP, or Gross Domestic Product, serves as a crucial economic measure used here to express the impact of costs due to Social Security and Medicare. The percentage of GDP indicates the economic weight these programs carry. Initially, Social Security claims a higher percentage but grows slower compared to Medicare:

• Social Security grows at 0.04% per year
• Medicare grows at 0.17% per year

By 2028, both costs will settle at 6.60% of GDP. Post-2028, Medicare's faster rise extends beyond, indicating a larger economic burden as its percentage grows swiftly. This highlights how the health of a country’s economy can be discussed through these percentages, offering insights into fiscal challenges such as budgetary planning and resource allocation.