From 1970 to 2000 , the percentage of Americans over the age of 25 that completed less than 12 years of school declined at the rate of about 0.5% per year. The percentage was 21.3% in 1970. Write a linear function describing the percentage in terms of years after 1970. The percentage reached 7.85 in 2010. When did your function predict this percentage would be reached? Write a linear function describing the percentage in terms of years after 1970 . Let the function P represent the percentage of Americans over the age of 25 that completed less than 12 years of school, and the variable X represent the number of years after 1970.

To solve this, let's start by setting up a linear function. We'll represent the percentage \( P \) of Americans over the age of 25 that completed less than 12 years of school as a linear function of time, with \( x \) representing the number of years after 1970.

### Step 1: Determine the slope of the linear function

From 1970 to 2000, the percentage of Americans completing less than 12 years of school declined at a rate of 0.5% per year. This means the slope of the line is \( -0.5 \) because the percentage is decreasing by 0.5% every year.

### Step 2: Write the initial condition

In 1970 (which is \( x = 0 \) since it's the starting point), the percentage was 21.3%. Therefore, we have the initial condition:
\[
P(0) = 21.3
\]

Using the slope-intercept form of a linear equation:
\[
P(x) = mx + b
\]
Where \( m \) is the slope and \( b \) is the y-intercept, which in this case is the initial percentage in 1970.

### Step 3: Substitute known values

We know:
- Slope \( m = -0.5 \)
- Initial percentage \( P(0) = 21.3 \), which means \( b = 21.3 \)

Thus, the equation becomes:
\[
P(x) = -0.5x + 21.3
\]

### Step 4: Find when the percentage reaches 7.85%

We are asked when the percentage reaches 7.85%. To find this, set \( P(x) = 7.85 \) and solve for \( x \).

\[
7.85 = -0.5x + 21.3
\]

Subtract 21.3 from both sides:
\[
7.85 - 21.3 = -0.5x
\]
\[
-13.45 = -0.5x
\]

Divide by -0.5:
\[
x = \frac{-13.45}{-0.5} = 26.9
\]

### Step 5: Interpretation

This means the percentage of Americans over 25 who completed less than 12 years of school would reach 7.85% about 26.9 years after 1970. So, according to the function, this percentage would be reached around:
\[
1970 + 26.9 \approx 1997
\]

Thus, according to the linear function, the percentage would have reached 7.85% in 1997, even though it actually reached this value in 2010. This discrepancy suggests that the rate of decline might have slowed down after 2000.

### Final Linear Function

The linear function that describes the percentage of Americans over 25 who completed less than 12 years of school in terms of years after 1970 is:
\[
P(x) = -0.5x + 21.3
\]

Would you like more details on any part of the solution or have any other questions?

Here are five related questions for further exploration:
1. What would the percentage be in the year 1985 according to the function?
2. What does the slope of the linear function represent in real-world terms?
3. How would the function change if the rate of decline was 0.3% per year instead?
4. How would you adjust the function if you wanted to model the decline continuing until 2020?
5. What assumptions does this linear model make about the future trend of the percentage?

**Tip:** When interpreting linear models, remember the slope represents the rate of change, while the y-intercept provides the starting value at the reference point (here, 1970).

From 1970 to 2000, the percentage of Americans over the age of 25 that completed less than 12 years of school declined at the rate of about 0.5% per year. The percentage was 21.3% in 1970. Write a linear function describing the percentage in terms of years after 1970. The percentage reached 7.85 in 2010. When did your function predict this percentage would be reached? Write a linear function describing the percentage in terms of years after 1970. Let the function P represent the percentage of Americans over the age of 25 that completed less than 12 years of school, and the variable X represent the number of years after 1970.

Explore how to model the decline in the percentage of Americans over the age of 25 who completed less than 12 years of school from 1970 to 2000 using a linear function. Learn to calculate when the percentage reached 7.85% based on a rate of 0.5% per year, starting from 21.3% in 1970. Suitable for students in Grades 9-12.

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