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.