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Chapter 1: Problem 17

Social Security According to the Social Security Advisory Board, the number of workers per beneficiary of the Social Security program was 3.3 in 1996 and is projected to decline by \(1.46 \%\) each year through \(2030 .\) a. Find a model for the number of workers per beneficiary from 1996 through 2030 . b. What does the model predict the number of workers per beneficiary will be in \(2030 ?\)

Short Answer

Expert verified
The number of workers per beneficiary in 2030 is predicted to be approximately 1.98.

Step by step solution

01

Define the Initial Value

Let's set the number of workers per beneficiary in 1996 as the initial value, which is given as 3.3. This will be the starting point for our model.
02

Determine the Rate of Decrease

The problem states that the number of workers per beneficiary declines by \(1.46\%\) each year. Express this percentage as a decimal by dividing by 100, resulting in 0.0146.
03

Create the Exponential Model

We can model the number of workers per beneficiary as an exponential decay function. The general form of an exponential decay model is \(y = a(1 - r)^t\), where \(a\) is the initial value, \(r\) is the decay rate, and \(t\) is the time in years from the starting point. For this problem, \(a = 3.3\) and \(r = 0.0146\). Thus, the model is \(y = 3.3(1 - 0.0146)^t\).
04

Calculate for the Year 2030

To find the predicted number of workers per beneficiary in 2030, calculate \(t\) as the difference between 2030 and 1996, which is 34 years. Substituting \(t = 34\) into our model, we get: \[y = 3.3(1 - 0.0146)^{34}\].
05

Solve the Exponential Equation

Calculate the value of \(y\) using the model from Step 4. First, compute the base: \((1 - 0.0146) = 0.9854\). Then, raise this base to the 34th power and multiply by 3.3. Use a calculator to find: \[y = 3.3(0.9854)^{34} \approx 3.3 \times 0.5987 \approx 1.9757\]. This predicts the number of workers per beneficiary in 2030.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Model
In mathematics, an exponential model is used to describe situations where quantities change at rates proportional to their current size. This type of model is particularly useful for representing population growth or decay, where change occurs continuously and compounds over time. In our context, the exponential decay model is handy to predict the number of workers per beneficiary in the Social Security program.

The formula for exponential decay is simple and intuitive:
  • \( y = a(1 - r)^t \)
Here:
  • \( y \) is the amount you're solving for, which would be the number of workers per beneficiary in any given future year.
  • \( a \) is the initial value. For our problem, this is the 3.3 workers per beneficiary in 1996.
  • \( r \) is the rate of decrease expressed as a decimal.
  • \( t \) is the time in years from the starting point.
This model gives a clear understanding of how a consistent rate of change affects the situation over time.
Rate of Decrease
The rate of decrease is a critical aspect of any exponential decay model. It signifies how quickly or slowly a quantity diminishes over time.

The more significant the rate, the faster the decrease. In the Social Security projections example, this rate is given as \(1.46\%\) per year. To incorporate this into calculations, it鈥檚 essential to convert the percentage into a decimal:
  • \(1.46\% \rightarrow 0.0146\).
Using this rate in the model allows us to effectively chart the diminishing number of workers per beneficiary. This conversion is critical because calculations occur in decimal form. Therefore, understanding the rate of change and correctly converting it is vital to applying the exponential model accurately.
Social Security Projections
Social Security projections rely heavily on data-backed models to predict future trends and scenarios. For instance, projecting the number of workers per beneficiary involves accounting for various factors including the documented rate of decrease.

This particular projection suggests a steady decline in the worker-to-beneficiary ratio from 1996 to 2030. This implies potential impacts on the sustainability of the Social Security program, as fewer workers may result in fewer contributions to the system, adding stress to the payout infrastructure.

Predictive modeling, like the one used in this exercise, is invaluable:
  • It helps policymakers prepare for future challenges.
  • Identifies trends early, allowing time to implement solutions.
  • Provides insight, enabling informed decision-making.
Thus, this model is more than an equation 鈥� it鈥檚 a tool for foresight and planning in social welfare governance.

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Most popular questions from this chapter

The following two functions have a common input, year \(y: D\) gives the total number of debit-card transactions, and \(P\) gives the number of point-of-sale debitcard transactions. a. Using function notation, show how to combine the two functions to create a third function, \(r\), that gives the percentage of debit-card transactions that were conducted at the point of sale in year \(y\) b. What are the output units of the new function?

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Stolen Bases San Francisco Giants legend Willie Mays's cumulative numbers of stolen bases between 1951 and 1963 are as shown below. Bases Stolen by Willie Mays (cumulative) \begin{tabular}{|c|c|c|c|} \hline Year & Stolen Bases & Year & Stolen Bases \\ \hline 1951 & 7 & 1958 & 152 \\ \hline 1952 & 11 & 1959 & 179 \\ \hline 1953 & 11 & 1960 & 204 \\ \hline 1954 & 19 & 1961 & 222 \\ \hline 1955 & 43 & 1962 & 240 \\ \hline 1956 & 83 & 1963 & 248 \\ \hline 1957 & 121 & & \\ \hline \end{tabular} (Source: www.baschall-reference.com) a. Find a logistic model for the data with input data aligned so that \(t=0\) in \(1950 .\) b. According to the model, how many bases did Mays steal in 1964 ? c. In 1964 Mays stole 19 bases. Does the model overestimate or underestimate the actual number? By how much?

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