To graph the relationship between the amount of fuel remaining in Karl's truck and the distance driven, we need to establish a few key variables:
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Initial Fuel Amount: Let's denote the initial amount of fuel in liters as \( F \) (for example, this could be any specific number you have, such as 100 liters).
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Fuel Consumption: Let’s denote the fuel consumption rate as \( C \) liters per kilometer (for example, this could also be a specific number like 0.05 liters/km).
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Distance Driven: We will denote the distance driven in kilometers as \( D \).
The relationship between the fuel remaining in the truck and the distance driven can be expressed mathematically. The remaining fuel \( R \) (in liters) after driving \( D \) kilometers can be calculated with the following equation:
\[ R = F - (C \times D) \]
Where:
- \( R \) is the remaining fuel in liters,
- \( F \) is the initial amount of fuel,
- \( C \) is the fuel consumption rate in liters per kilometer,
- \( D \) is the distance driven in kilometers.
Steps to Graph the Relationship
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Identify values for \( F \) and \( C \):
- Let’s say \( F = 100 \) liters.
- Let’s say \( C = 0.05 \) liters/kilometer.
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Create a table of values for \( D \) (distance driven) and calculate \( R \) (remaining fuel):
| Distance Driven (km) | Remaining Fuel (liters) | |----------------------|--------------------------| | 0 | \( 100 - (0.05 \times 0) = 100 \) | | 10 | \( 100 - (0.05 \times 10) = 99.5 \) | | 20 | \( 100 - (0.05 \times 20) = 99 \) | | 30 | \( 100 - (0.05 \times 30) = 98.5 \) | | 40 | \( 100 - (0.05 \times 40) = 98 \) | | 50 | \( 100 - (0.05 \times 50) = 97.5 \) | | 60 | \( 100 - (0.05 \times 60) = 97 \) | | 70 | \( 100 - (0.05 \times 70) = 96.5 \) | | 80 | \( 100 - (0.05 \times 80) = 96 \) | | 90 | \( 100 - (0.05 \times 90) = 95.5 \) | | 100 | \( 100 - (0.05 \times 100) = 95 \) |
- Plot these values on a graph:
- The x-axis represents Distance Driven (D) in kilometers.
- The y-axis represents Remaining Fuel (R) in liters.
Graph Interpretation
- Linear Relationship: The relationship is linear, with a negative slope, since more distance driven leads to less fuel remaining.
- Intercept: The y-intercept will be at \( (0, F) \) which in this case is \( 100 \) liters when no distance is driven.
- End Point: If you keep driving until the fuel runs out, you can find the maximum distance \( D_{\text{max}} \) that can be driven by setting \( R = 0 \):
\[ 0 = F - (C \times D_{\text{max}}) \Rightarrow D_{\text{max}} = \frac{F}{C} \] In this case for \( F = 100 \) and \( C = 0.05 \): \[ D_{\text{max}} = \frac{100}{0.05} = 2000 \text{ km} \]
This means after \( 2000 \) kilometers, Karl will have exactly \( 0 \) liters of fuel left.
You can now create a line graph based on the data.
Conclusion
From the table and graph, it becomes clear how fuel consumption affects distance driven, and you can visually assess how much fuel remains as the distance increases. Adjust \( F \) and \( C \) as needed for different scenarios.