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The given problem involves a mathematical model for public high school enrollment in the U.S. from 1986 to 1996. Let's break down the task step by step.

### The model:
The equation provided in the image is:
\[
N = 0.05t^2 - 0.42t + 12.33
\]
Where:
- \(N\) is the number of students enrolled in U.S. public high schools (in millions).
- \(t\) is the time in years since 1986, so \(t = 0\) represents the year 1986, \(t = 1\) represents 1987, and so on.

Now, let's address each part of the problem:

### (a) Use functional notation to express the number of students enrolled in 1989, and calculate that value.
For 1989, \(t = 1989 - 1986 = 3\). We need to substitute \(t = 3\) into the model equation to find \(N(3)\).

\[
N(3) = 0.05(3)^2 - 0.42(3) + 12.33
\]
Let's compute that.

\[
N(3) = 0.05(9) - 0.42(3) + 12.33
\]
\[
N(3) = 0.45 - 1.26 + 12.33
\]
\[
N(3) = 11.52
\]
Thus, the number of students enrolled in U.S. public high schools in 1989 was approximately **11.52 million**.

### (b) Explain in practical terms what \(N(3)\) means and calculate that value.
We already calculated \(N(3)\), which is 11.52. In practical terms, \(N(3)\) represents the estimated number of students enrolled in U.S. public high schools in the year 1989, based on the provided model. The model predicts that around 11.52 million students were enrolled in public high schools in 1989.

### (c) In what year was the enrollment the smallest?
To find when enrollment was smallest, we need to find the minimum of the quadratic function \(N(t)\). The general form of a quadratic equation is \(N(t) = at^2 + bt + c\), and the minimum value of a parabola (if it opens upwards) occurs at:
\[
t = \frac{-b}{2a}
\]
In this case, \(a = 0.05\) and \(b = -0.42\), so we can find \(t\):

\[
t = \frac{-(-0.42)}{2(0.05)} = \frac{0.42}{0.1} = 4.2
\]

So, the minimum enrollment occurs when \(t = 4.2\), which corresponds to the year:
\[
1986 + 4.2 = 1990.2
\]
Thus, the enrollment was smallest around the year **1990**.

---

Would you like me to explain any specific part further, or dive deeper into any concept? Here are five related questions to deepen your understanding:

1. How would the enrollment change if the model were extended to 2000?
2. What is the significance of the coefficients 0.05, -0.42, and 12.33 in the model?
3. How can you determine if a quadratic function has a minimum or maximum?
4. How would you interpret the model if it predicted a negative enrollment?
5. Could there be other factors affecting enrollment that aren't captured in this quadratic model?

**Tip:** When dealing with quadratic functions, remember that the sign of the coefficient of \(t^2\) (positive or negative) determines whether the parabola opens upward (minimum) or downward (maximum).

One model for the number of students enrolled in U.S. public high schools as a function of time since 1986 is N = 0.05t^2 - 0.42t + 12.33. Here N is the enrollment in millions of students, t is the time in years since 1986, and the model is relevant from 1986 to 1996. (a) Use functional notation to express the number of students enrolled in U.S. public high schools in the year 1989, and then calculate that value. (b) Explain in practical terms what N(3) means and calculate that value. (c) In what year was the enrollment the smallest?

Explore how to calculate the enrollment in U.S. public high schools using a quadratic model from 1986 to 1996. Learn how to find the number of students in 1989 and determine when the enrollment was at its smallest using algebraic methods. Suitable for Grades 9-11.

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