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(6) The percentage of Americans who are retired has been increasing over the last several decades. This is causing some concern because health care, Social Security, and other costs will be the responsibility of a smaller group of people. That is, as the percentage of retired people increases, the percentage of working-age people decreases.

The following model predicts the percentage of retired people based on demographic data:
[tex]\[ R = \frac{t}{588.24} - 3.24 \][/tex]
where [tex]\( R \)[/tex] is the percentage (as a decimal) of Americans who are retired in the year [tex]\( t \)[/tex] (for example, in 2016, [tex]\( t = 2016 \)[/tex]).



Answer :

GinnyAnswer
Certainly! Let's solve this problem step by step.

### Problem Statement
We are given a mathematical model that predicts the percentage, [tex]\( R \)[/tex], of Americans who are retired in a given year [tex]\( t \)[/tex]. The model is expressed by the formula:
[tex]\[ R = \frac{t}{588.24} - 3.24 \][/tex]
Here, [tex]\( R \)[/tex] is the percentage of retired people (expressed as a decimal), and [tex]\( t \)[/tex] is the year.

### Given Year: 2016
We are specifically asked about the year 2016. In this context, [tex]\( t = 2016 \)[/tex].

### Step-by-Step Solution:

1. Substitute [tex]\( t \)[/tex] with 2016 in the equation:
[tex]\[ R = \frac{2016}{588.24} - 3.24 \][/tex]

2. Perform the division:
Calculate [tex]\( \frac{2016}{588.24} \)[/tex]:
[tex]\[ \frac{2016}{588.24} \approx 3.426 \][/tex]

3. Subtract 3.24 from the result:
[tex]\[ R = 3.426 - 3.24 \][/tex]

4. Complete the subtraction:
[tex]\[ R = 0.186 \][/tex]

### Final Answer
When [tex]\( t = 2016 \)[/tex], the percentage of Americans who are retired, according to the model, is approximately [tex]\( 0.186 \)[/tex] or 18.6%.

To summarize:
In the year 2016, the model predicts that approximately 18.6% of Americans are retired.

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