From the image, the problem involves creating a linear equation to model the number of immigrants admitted to a country. Here’s how we can approach the problem:

Given:

• In 1950, the number of immigrants was 246,046.
• In 2003, the number of immigrants was 1,100,712.

(a) Write an equation assuming a linear change in the number of immigrants over time.

We need to write the equation of a line in slope-intercept form: $y = mx + b$ Where:

• $y$ is the number of immigrants.
• $x$ is the number of years since 1900 (for easier calculations).
• $m$ is the slope (rate of change in the number of immigrants per year).
• $b$ is the y-intercept (the number of immigrants in 1900).

Step 1: Calculate the slope $m$

The formula for slope between two points is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Here, the two points are:

• $(1950, 246046)$
• $(2003, 1100712)$

We convert the years into years since 1900:

• $x_1 = 1950 - 1900 = 50$
• $x_2 = 2003 - 1900 = 103$

Now calculate the slope: $m = \frac{1100712 - 246046}{103 - 50} = \frac{854666}{53} \approx 16125.79$

Step 2: Use point-slope form to find the equation

We can use one of the points to find the y-intercept $b$. Let's use the point (1950, 246046): $y = mx + b \Rightarrow 246046 = 16125.79(50) + b$ $246046 = 806289.5 + b$ $b = 246046 - 806289.5 = -560243.5$

Final Equation:

The equation for the number of immigrants is approximately: $y = 16125.79x - 560243.5$ where $x$ is the number of years since 1900.

(b) Predict the number of immigrants in 2013

For $x = 2013 - 1900 = 113$: $y = 16125.79(113) - 560243.5 = 1812184.67 - 560243.5 = 1251941.17$ Thus, approximately 1,251,941 immigrants would be admitted in 2013.

(c) Validity of the model

You would need to discuss whether the linear model remains valid for the entire century. The model assumes constant change, but immigration can fluctuate due to policies, wars, or economic factors, suggesting it may not hold over long periods.

Would you like further clarification or additional steps? Here are some related questions:

1. How can you calculate the rate of change in a different year range?
2. How do you interpret the y-intercept in this context?
3. Can a linear model be replaced with a different type of function (e.g., quadratic)?
4. What are some real-world factors that could affect this linear model?
5. How would you check if the predictions are accurate with actual data?

Tip: Always consider whether a linear model is reasonable when predicting future trends. In real-world data, trends might be non-linear!