Let's work through the tasks step by step based on the graph.

1. Story Problem:

A machine produces 360 widgets per shift. However, due to some technical issues, the machine’s output starts decreasing at a constant rate, as shown in the graph. After 6 hours, the machine produces 0 widgets. How many widgets does the machine produce each hour before it stops?

2. Table with Data from the Graph:

The data points can be read off the graph. Here is a table based on the visible points:

Here, "Hours" is the time since the machine started, and "Widgets Produced" is the number of widgets produced at that time.

3. Equation of the Line:

The line in the graph is linear, and we can use two points to find the equation.

Using the points $(0, 360)$ and $(6, 0)$:

1. Slope (m): $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 360}{6 - 0} = \frac{-360}{6} = -60$

2. Equation of the line (y = mx + b):

   • The slope is $-60$.
   • The y-intercept (b) is 360 (since $y = 360$ when $x = 0$).

   So, the equation of the line is: $y = -60x + 360$

4. Interpretation of the Slope, Y-Intercept, and X-Intercept:

• Slope: The slope of $-60$ means that the machine's output decreases by 60 widgets per hour.
• Y-Intercept: The y-intercept (360) represents the initial number of widgets produced at the beginning (at hour 0).
• X-Intercept: The x-intercept is 6 (when $y = 0$), meaning that after 6 hours, the machine stops producing widgets entirely.

Solving the Story Problem:

The story problem asks how many widgets the machine produces each hour before it stops. From the slope $-60$, we can conclude that the machine produces 60 fewer widgets every hour.

Would you like more details or clarifications on any of these steps?

Here are five related questions to deepen understanding:

1. How would the equation change if the machine took 8 hours to stop producing widgets instead of 6?
2. What would the graph look like if the machine increased its output by 60 widgets per hour?
3. If the machine produced 180 widgets in the second hour, what would the new slope be?
4. Can you find the average production rate of the machine over the entire 6-hour period?
5. How would you describe the relationship between hours and production mathematically (e.g., linear, exponential)?

Tip: When interpreting linear graphs, always look for key points like the y-intercept and slope to understand the relationship between variables clearly.