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The table provided in the image shows the life expectancy data from different years, and the question asks for the year when life expectancy will reach 83 years, based on a linear model. To solve this, we can follow these steps:

### Step 1: Define Variables
- \( t = 0 \) corresponds to the year 2000.
- \( L(t) \) is the life expectancy in the year corresponding to \( t \).
- The goal is to determine the year when life expectancy \( L(t) = 83 \) years.

### Step 2: Gather Data Points
From the table, we can extract two points:
- In 2000 (\( t = 0 \)), life expectancy \( L(0) = 76.75 \) years.
- In 2019 (\( t = 19 \)), life expectancy \( L(19) = 78.87 \) years.

### Step 3: Find the Linear Equation
The linear model will be of the form \( L(t) = mt + b \), where \( m \) is the slope, and \( b \) is the intercept. We can calculate the slope using the formula:

\[
m = \frac{L(19) - L(0)}{19 - 0} = \frac{78.87 - 76.75}{19} = \frac{2.12}{19} \approx 0.1116
\]

Now, using the point \( t = 0 \), \( L(0) = 76.75 \), we can find \( b \):

\[
L(0) = 76.75 = 0.1116(0) + b \implies b = 76.75
\]

So, the equation for life expectancy is:

\[
L(t) = 0.1116t + 76.75
\]

### Step 4: Solve for \( t \) when \( L(t) = 83 \)
Set \( L(t) = 83 \):

\[
83 = 0.1116t + 76.75
\]

Solving for \( t \):

\[
83 - 76.75 = 0.1116t \implies 6.25 = 0.1116t \implies t = \frac{6.25}{0.1116} \approx 56
\]

So, \( t = 56 \), which corresponds to the year:

\[
2000 + 56 = 2056
\]

Thus, the model predicts that life expectancy will reach 83 years in **2056** (Option C).

Would you like more details or have any questions? Here are five related questions:

1. How is a linear regression model constructed from data points?
2. How do you calculate the slope in a linear model?
3. What are the limitations of using a linear model for future predictions?
4. How do different data points affect the slope and intercept in linear regression?
5. How can we test the accuracy of the model as new data points emerge?

**Tip:** Always check if the data suggests a linear trend before applying a linear model for predictions.

The table gives values of average life expectancy, in years, for a child born in a given year in the United States. A linear regression is used to construct a linear function model L, where t represents the birth year, t = 0 is the year 2000, and L(t) represents the life expectancy in years. For what year does the model predict that the life expectancy of a child born in that year will be 83 years?

Discover how to predict life expectancy using linear regression. This problem involves calculating the slope and intercept of a linear equation based on life expectancy data from 2000 to 2019. Suitable for students in Grades 10-12, it highlights essential concepts in algebra and statistics.

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