f(x) = 500 + 33x where x is the number of years since 2015 and f(x) is measured in thousands of tons … find the slope and the rate at which the amount of ore produced changed in thousands of tons per year

The function given is:

\[
f(x) = 500 + 33x
\]

where:
- \( x \) is the number of years since 2015.
- \( f(x) \) is the amount of ore produced, measured in thousands of tons.

### 1. Identifying the Slope

The function \( f(x) = 500 + 33x \) is a linear equation in the slope-intercept form:

\[
f(x) = mx + b
\]

where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept (the value of \( f(x) \) when \( x = 0 \)).

Comparing this to \( f(x) = 500 + 33x \), we see that:

\[
m = 33
\]

### 2. Interpreting the Slope

The slope \( m = 33 \) represents the **rate of change** of the amount of ore produced with respect to time (in thousands of tons per year).

### 3. Rate of Change

The slope tells us that the amount of ore produced increases by **33 thousand tons per year**. 

### Conclusion

- **Slope:** 33
- **Rate of Change:** The amount of ore produced changes at a rate of 33 thousand tons per year.

Would you like a deeper explanation, or do you have any other questions?

### Additional Questions:
1. What is the ore production in 2020?
2. How much ore will be produced 10 years after 2015?
3. How would the equation change if the production rate was 40 thousand tons per year instead?
4. What would the slope represent if the equation modeled a decrease in production?
5. Can the rate of change be different at different points for a linear function?

**Tip:** The slope in a linear function always represents the

Discover how to find the slope and rate of change in ore production using the linear equation f(x) = 500 + 33x, where x represents the number of years since 2015, and f(x) is in thousands of tons. This problem explores key algebra concepts and is suitable for students in Grades 9-11.

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