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Let's analyze and answer each part of the problem.

Given:
\[ T(x) = 150x + 32,000 \]
where \( x \) is the number of years since 1996, and \( T(x) \) is the amount of trash in thousands of tons.

### (a) Sketch a graph of \( T \).

To sketch the graph, we note that \( T(x) = 150x + 32,000 \) represents a linear function in slope-intercept form, \( y = mx + b \), where:
- \( m = 150 \) (the slope),
- \( b = 32,000 \) (the y-intercept).

1. **Intercepts**:
   - **y-intercept**: At \( x = 0 \), \( T(0) = 32,000 \), so the graph crosses the y-axis at \( (0, 32,000) \).
   - **Slope**: For every increase of 1 in \( x \), \( T(x) \) increases by 150, indicating a steady upward trend.

The graph is a straight line starting from \( (0, 32,000) \) and increasing linearly with a slope of 150.

### (b) What is the slope of the graph?

The slope of the graph is given directly by the coefficient of \( x \) in the equation:
\[
\text{slope} = 150
\]

### (c) At what rate is the amount of trash in the landfill increasing per year?

The rate at which the amount of trash increases per year is represented by the slope, which is 150. Since \( T(x) \) is measured in thousands of tons, the trash increases by:
\[
150 \text{ thousand tons per year, or } 150,000 \text{ tons per year}.
\]

Would you like me to plot the graph for you? Also, let me know if you want a more detailed explanation of any part.

Here are five related questions for further exploration:

1. What would the equation look like if the rate of trash increase changed to 200 thousand tons per year?
2. How much trash would be in the landfill after 10 years (i.e., by 2006)?
3. How would the graph change if the initial amount of trash were 40,000 thousand tons?
4. What does a negative slope imply in the context of this landfill model?
5. How would you modify the function \( T(x) \) if trash were removed from the landfill at a constant rate each year?

**Tip:** In linear equations, the slope directly represents the rate of change, making it useful in predicting trends over time.

The amount of trash in a county landfill is modeled by the function T(x) = 150x + 32,000, where x is the number of years since 1996 and T(x) is measured in thousands of tons. (a) Sketch a graph of T. (b) What is the slope of the graph? (c) At what rate is the amount of trash in the landfill increasing per year?

Explore how to analyze the function T(x) = 150x + 32,000, which models the increase in trash in a county landfill. Learn how to sketch the graph, identify the slope, and determine the rate of increase per year. This problem involves linear functions and is suitable for high school students.

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