To answer your questions, we need to determine the relationship between the distance traveled and the number of gallons of fuel used based on the information you've provided.
From the information given, it sounds like you're dealing with a direct variation: the distance traveled (y) varies directly with the number of gallons (x) used.
(a) To find how many miles the train travels per gallon, we would calculate the slope (rise/run) using the data points provided in the graph.
Let's assume that you have points that represent the distance traveled (y) and gallons used (x). For instance, if you have a point at (x = 50, y = 400), then:
- When the train uses 50 gallons, it travels 400 miles.
- Then you can calculate the miles per gallon: \[ \text{Miles per gallon} = \frac{\text{Distance (y)}}{\text{Gallons (x)}} = \frac{400}{50} = 8 \text{ miles per gallon.} \]
From the direct variation, you can see that the slope of the line on the graph will also give you this same value.
(b) The slope of the graph (m) can be found using any two points on the line. The slope formula is: \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}. \]
Using the same example points (let’s assume two points: (50, 400) and (100, 800)):
- \( y_1 = 400, y_2 = 800, x_1 = 50, x_2 = 100 \)
- Now using the slope formula: \[ m = \frac{800 - 400}{100 - 50} = \frac{400}{50} = 8. \]
Thus, the answers would be:
(a) The train travels 8 miles per gallon. (b) The slope of the graph is 8.
If you have specific points from the graph, please provide them, and I can give you more tailored answers!