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Problem Summary:

1. In 1940, there were 241,595 immigrants admitted.
2. In 2008, there were 1,124,742 immigrants admitted.
3. You are asked to:
   • (a) Write an equation expressing the number of immigrants $y$ as a linear function of $t$, the number of years after 1900.
   • (b) Use this equation to predict the number of immigrants in 2017.
   • (c) Analyze the validity of using this model across the entire 20th century.

(a) Writing the Linear Equation:

Let $t$ be the number of years after 1900, and $y(t)$ be the number of immigrants admitted in a year.

We are given two points on the line:

• In 1940 ($t = 40$), the number of immigrants is $y(40) = 241,595$.
• In 2008 ($t = 108$), the number of immigrants is $y(108) = 1,124,742$.

We can use the point-slope form of the linear equation: $y = mt + b$ Where $m$ is the slope, and $b$ is the y-intercept.

1. Find the slope $m$: $m = \frac{y_2 - y_1}{t_2 - t_1} = \frac{1,124,742 - 241,595}{108 - 40} = \frac{883,147}{68} = 12,986$

2. Find the y-intercept $b$: Now, using one of the points (let’s use 1940: $t = 40, y = 241,595$): $y = mt + b \implies 241,595 = 12,986(40) + b$ $241,595 = 519,440 + b \implies b = 241,595 - 519,440 = -277,845$

Thus, the linear equation is: $y(t) = 12,986t - 277,845$

(b) Predicting the Number of Immigrants in 2017:

For $t = 2017 - 1900 = 117$, we can substitute $t = 117$ into the equation: $y(117) = 12,986(117) - 277,845 = 1,520,562 - 277,845 = 1,242,717$ So, the predicted number of immigrants in 2017 is 1,242,717.

(c) Discussing the Validity of the Model:

The y-intercept $b = -277,845$ suggests that, according to the model, immigration numbers would have been negative in the early 1900s. This is clearly not possible, which indicates that the model may not be valid across the entire 20th century. The assumption of a constant linear growth is likely too simplistic, as immigration rates fluctuate due to various historical, political, and economic factors. Therefore, while the linear model may work for the given period (1940–2008), it would not accurately represent the entire 20th century.

Would you like any further details or clarification?

Related Questions:

1. How does changing the time frame affect the slope of a linear model like this one?
2. How would you modify the model to account for non-linear growth in immigration?
3. What factors might cause the number of immigrants to change more drastically in certain years?
4. Could another type of function (e.g., exponential) be a better fit for modeling immigration? Why or why not?
5. How could historical events be incorporated into this model to improve accuracy?

Tip:

When modeling real-world data, it’s important to consider external factors that may influence the data over time. A linear model is a simplification and might not always fit well across large time scales.