In a particular country the number of retirees was approximately 150 per thousand people aged 20-64 in the first year of keeping that data. Forty years later, this number rose to approximately 200, and it was projected to rise to 275 after 50 years. Model N as a piecewise linear function of the time t in years since the first year, letting t = 0 represent the first year of keeping data. Then use your model to project the number of retires per thousand people aged 20-64 in year 47. (Round your answer to the nearest integer.)

We are asked to model \( N(t) \), the number of retirees per thousand people aged 20-64, as a piecewise linear function of \( t \), where \( t = 0 \) represents the first year of data collection. The data points provided are:

- At \( t = 0 \) (the first year), \( N(0) = 150 \).
- At \( t = 40 \), \( N(40) = 200 \).
- At \( t = 50 \), \( N(50) = 275 \).

To model \( N(t) \), we can split the problem into two linear segments:

1. The first segment covers \( 0 \leq t \leq 40 \).
2. The second segment covers \( 40 \leq t \leq 50 \).

### Step 1: Equation of the first line for \( 0 \leq t \leq 40 \)
We need to find the equation of the line passing through the points \( (0, 150) \) and \( (40, 200) \).

The slope \( m_1 \) of the line is given by:
\[
m_1 = \frac{200 - 150}{40 - 0} = \frac{50}{40} = 1.25
\]

Using the point-slope form of a linear equation:
\[
N(t) = m_1 t + N(0)
\]
\[
N(t) = 1.25 t + 150 \quad \text{for} \quad 0 \leq t \leq 40
\]

### Step 2: Equation of the second line for \( 40 \leq t \leq 50 \)
We now find the equation of the line passing through \( (40, 200) \) and \( (50, 275) \).

The slope \( m_2 \) of this line is:
\[
m_2 = \frac{275 - 200}{50 - 40} = \frac{75}{10} = 7.5
\]

Again, using the point-slope form:
\[
N(t) = m_2 (t - 40) + 200
\]
\[
N(t) = 7.5 (t - 40) + 200 \quad \text{for} \quad 40 \leq t \leq 50
\]
\[
N(t) = 7.5 t - 300 + 200
\]
\[
N(t) = 7.5 t - 100 \quad \text{for} \quad 40 \leq t \leq 50
\]

### Step 3: Projection for year \( t = 47 \)
Since \( t = 47 \) falls within the second segment \( (40 \leq t \leq 50) \), we use the second equation:
\[
N(47) = 7.5 \times 47 - 100 = 352.5 - 100 = 252.5
\]
Rounding to the nearest integer:
\[
N(47) = 253
\]

Thus, the projected number of retirees per thousand people aged 20-64 in year 47 is \( \boxed{253} \).

Do you want further clarification or details on any steps? 

Here are 5 related questions:
1. How would the result change if the rate of increase between years 40 and 50 was slower?
2. What is the significance of using a piecewise linear function in modeling this data?
3. How could this model be extended beyond 50 years?
4. What assumptions underlie the use of a linear model in this context?
5. How does the choice of rounding affect the final result?

**Tip**: When modeling real-world data, always check if linear functions are appropriate for the entire domain or if non-linear models might better reflect the situation.

In a particular country the number of retirees was approximately 150 per thousand people aged 20-64 in the first year of keeping that data. Forty years later, this number rose to approximately 200, and it was projected to rise to 275 after 50 years. Model N as a piecewise linear function of the time t in years since the first year, letting t = 0 represent the first year of keeping data. Then use your model to project the number of retirees per thousand people aged 20-64 in year 47. (Round your answer to the nearest integer.)

Explore how to model the number of retirees per thousand people aged 20-64 using piecewise linear functions. The problem involves calculating projections based on data points from years 0, 40, and 50. Learn to solve this step-by-step using key algebraic concepts, slopes, and linear interpolation. Perfect for high school students learning algebra and piecewise functions.

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