Let n be the number of oil refineries at t years since 

20042004. 

A reasonable model of the number of oil refineries is 

nequals=minus−4.994.99tplus+165.34165.34.

**a.** What is the​ slope? What does it mean in this​ situation?

**b.** What is the​ n-intercept? What does it mean in this​ situation?

**c.** Predict the number of refineries in 

20162016.

We are given the linear model:

\[
n(t) = -4.99t + 165.34
\]

where:
- \( n(t) \) represents the number of oil refineries,
- \( t \) is the number of years since 2004.

Let's go through the parts of the question step by step.

### **a. What is the slope? What does it mean in this situation?**
The slope of the equation is the coefficient of \( t \), which is \( -4.99 \).

- **Slope**: \( m = -4.99 \)
- **Meaning**: The slope represents the rate of change in the number of oil refineries per year. Since the slope is negative, it indicates that the number of oil refineries is **decreasing** by 4.99 refineries per year.

### **b. What is the n-intercept? What does it mean in this situation?**
The \( n \)-intercept occurs when \( t = 0 \). To find this, we can evaluate the model at \( t = 0 \):

\[
n(0) = -4.99(0) + 165.34 = 165.34
\]

- **n-intercept**: \( 165.34 \)
- **Meaning**: This represents the number of oil refineries in the base year, 2004. So in 2004, there were approximately **165 refineries**.

### **c. Predict the number of refineries in 2016.**
To predict the number of refineries in 2016, we first find the value of \( t \) for 2016. Since \( t \) is the number of years since 2004, we have:

\[
t = 2016 - 2004 = 12
\]

Now, substitute \( t = 12 \) into the equation:

\[
n(12) = -4.99(12) + 165.34 = -59.88 + 165.34 = 105.46
\]

So, the predicted number of oil refineries in 2016 is approximately **105 refineries**.

### Summary:
- **a. Slope**: The slope is \( -4.99 \), which means the number of refineries decreases by 4.99 per year.
- **b. n-intercept**: The n-intercept is \( 165.34 \), meaning there were about 165 refineries in 2004.
- **c. Prediction for 2016**: There were approximately 105 refineries predicted for 2016.

Would you like any more details, or do you have any questions?

Here are 5 related questions you can consider:
1. How would the interpretation change if the slope were positive?
2. What would the model predict for the number of refineries in 2025?
3. What does the equation suggest about the long-term trend in oil refineries?
4. How can we interpret the year when there are predicted to be zero refineries?
5. What other factors might affect the number of refineries that this model doesn't account for?

**Tip**: Always interpret the slope in terms of the rate of change in real-world applications, as it gives insight into the behavior of the modeled situation.

Let n be the number of oil refineries at t years since 2004. A reasonable model of the number of oil refineries is n = -4.99t + 165.34.

a. What is the slope? What does it mean in this situation?
b. What is the n-intercept? What does it mean in this situation?
c. Predict the number of refineries in 2016.

Explore how to model the number of oil refineries over time using the linear equation n = -4.99t + 165.34. Understand the significance of the slope and intercept in this context, and predict the number of refineries in 2016. This problem covers key algebraic concepts suitable for Grades 9-12.

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