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Let's break down and solve the question presented in the image.

The problem states:
The amount of copper ore produced from a copper mine in Arizona is modeled by the function:
\[
f(x) = 200 + 32x
\]
where \( x \) is the number of years since 2005, and \( f(x) \) is measured in thousands of tons.

The questions are:

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

The function \( f(x) = 200 + 32x \) is in the form of a linear equation, \( y = mx + b \), where \( m \) represents the slope of the line.

For this function:
- \( m = 32 \), so the **slope is 32**.

### (b) At what rate is the amount of ore produced changing?

The slope, \( m = 32 \), represents the rate of change of \( f(x) \) with respect to \( x \). Since \( f(x) \) is measured in thousands of tons and \( x \) is the number of years, this slope indicates that the amount of ore produced is increasing by **32 thousand tons per year**.

To summarize:
1. **The slope of the graph is 32.**
2. **The rate of change of the amount of ore produced is 32 thousand tons per year.**

Would you like more details on how these linear functions model such real-world situations? 

### Related Questions
1. What does the y-intercept (200) represent in this context?
2. How would the model change if the production rate increased to 40 thousand tons per year?
3. If production started in 2005, what would the total production be in the year 2025?
4. How would you represent a decrease in ore production instead of an increase?
5. Can we use this model to predict when the production will reach 1 million tons?

#### Tip:
In linear models, the slope represents the rate of change, and the y-intercept often represents the starting point or initial condition in real-world scenarios.

The amount of copper ore produced from a copper mine in Arizona is modeled by the function f(x) = 200 + 32x where x is the number of years since 2005, and f(x) is measured in thousands of tons. (a) What is the slope of the graph? (b) At what rate is the amount of ore produced changing?

Explore the analysis of copper ore production from a mine in Arizona using the function f(x) = 200 + 32x. This linear model explains the slope of the graph and the rate of change in production over time. Suitable for high school students in Grades 9-10.

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