To solve this problem, we need to model the country's annual egg production using a linear function based on the provided data points. The data points are:

• (1995, 51.7)
• (1999, 60.4)

We will let $x$ represent the year, where $x = 0$ corresponds to 1995, $x = 1$ to 1996, and so on. $y$ represents the egg production in billions.

Step 1: Determine the linear function

A linear function has the form $y = mx + b$, where:

• $m$ is the slope.
• $b$ is the y-intercept.

Calculate the slope $m$:

The slope is calculated using the two given points:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the points (1995, 51.7) and (1999, 60.4) into the formula. We can convert the years to $x$ values as follows:

• $x_1 = 0$ (for 1995), $y_1 = 51.7$
• $x_2 = 4$ (for 1999), $y_2 = 60.4$

$m = \frac{60.4 - 51.7}{4 - 0} = \frac{8.7}{4} = 2.175$

Determine the y-intercept $b$:

Using the slope $m = 2.175$ and one of the points, we can find $b$. We'll use the point (0, 51.7):

$y = mx + b \quad \Rightarrow \quad 51.7 = 2.175(0) + b$

$b = 51.7$

Step 2: Write the linear equation

The linear model for the egg production is:

$y = 2.175x + 51.7$

Step 3: Predict egg production in 2010 and 2021

For 2010 (which corresponds to $x = 15$):

$y = 2.175(15) + 51.7 = 32.625 + 51.7 = 84.325 \text{ billion eggs}$

For 2021 (which corresponds to $x = 26$):

$y = 2.175(26) + 51.7 = 56.55 + 51.7 = 108.25 \text{ billion eggs}$

Conclusion

• The predicted egg production in 2010 is approximately 84.3 billion.
• The predicted egg production in 2021 is approximately 108.3 billion.

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to expand on this topic:

1. How can you interpret the slope of the linear function in the context of this problem?
2. What assumptions are we making by using a linear model to predict egg production?
3. How would you verify the accuracy of this model using additional data points?
4. Can you calculate the rate of change in egg production between 1995 and 1999?
5. What are potential limitations of using a linear model for long-term predictions?

Tip: Always plot the data points and the linear model to visually inspect the fit before making predictions. This helps to assess whether a linear model is appropriate.