Let's go through the problem you've shared.

The problem provides the following data points for the number of immigrants admitted to a country:

• In 1950, there were 246,046 immigrants.
• In 2003, the number of immigrants increased to 1,100,712.

We are asked to solve several parts based on this information:

Part (a)

Linear equation: We need to assume that immigration trends follow a linear model and write an equation in slope-intercept form, $y = mx + b$, where $y$ represents the number of immigrants, and $t$ represents the number of years after 1900.

First, we calculate the slope $m$. The slope is the rate of change of the number of immigrants per year:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1,100,712 - 246,046}{2003 - 1950}$

After calculating the slope, we'll use one of the points (either 1950, 246,046 or 2003, 1,100,712) to solve for the y-intercept $b$.

Part (b)

Immigrants in 2013: We can predict the number of immigrants admitted in 2013 by using the equation derived from part (a). Set $t = 113$ (since 2013 is 113 years after 1900) and plug it into the equation to find the value of $y$.

Part (c)

Validity of the model: Here we will discuss whether the model is valid over the entire 20th century. Since this is a linear model, it assumes a constant rate of change, which may not account for various external factors like policy changes, world events, or population growth that can affect immigration rates.

Let me calculate the slope and the rest of the details for you.Here's the detailed breakdown:

Part (a)

The linear equation for the number of immigrants admitted over time, in slope-intercept form, is:

$y = 16,125.77t - 560,242.68$

Where:

• $t$ is the number of years after 1900.
• The slope $16,125.77$ represents the increase in immigrants per year.
• The y-intercept $-560,242.68$ suggests that the model would predict negative immigrants around the early 1900s, which raises questions about the model's validity in those years.

Part (b)

The predicted number of immigrants in 2013 is approximately:

$y \approx 1,261,970$

This is obtained by plugging $t = 113$ (2013 is 113 years after 1900) into the equation.

Part (c)

The validity of the model can be discussed in terms of its limitations. While the linear equation works well to estimate immigration between 1950 and 2003, it's likely not reliable for the entire 20th century. The y-intercept suggests a negative number of immigrants early in the century, which is unrealistic. Additionally, factors like economic, political changes, and migration policies are not accounted for in this model.

Would you like more details or have any questions? Here are some additional questions you can think about:

1. How would the model change if there were more data points?
2. What could cause the rate of immigration to increase or decrease faster than this model predicts?
3. Could a quadratic or exponential model better fit the immigration data over a longer period?
4. What are the real-world factors influencing immigration trends that aren't considered in this model?
5. How sensitive is the model to changes in the slope?

Tip: When using linear models, always consider the context and possible limitations, especially when extrapolating data far outside the range of given points.