To analyze the functions and their graphs:
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Hot Air Balloon: The function \( f(x) = -x^2 + 150 \) is a downward-opening parabola. At \( x = 0 \) (the starting point, representing the initial time), the height is 150 meters. Since it opens downward, the balloon will eventually start descending and will reach the ground (height = 0) at some point.
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Mine Shaft Elevator: The function \( g(x) = -20x \) is a straight line with a negative slope. At \( x = 0 \), the elevator starts at a height of 0 meters (ground level) and will descend as time increases.
Now, comparing the two scenarios:
- The hot air balloon starts at 150 meters and will eventually fall to the ground.
- The mine shaft elevator starts at ground level (0 meters) and descends.
Given these points, let's evaluate the statements:
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Negative range values don't make sense for either scenario.: This statement is true since negative heights would not make sense in the context of heights for both the balloon and the elevator.
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The balloon will reach the ground before the elevator car does.: This might not necessarily be true, as the balloon starts at 150 meters and will take time to fall, depending on the vertex of the parabola.
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The starting point of the balloon is higher than that of the elevator.: This statement is true; the balloon starts at 150 meters, while the elevator starts at 0 meters.
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Positive domain values don't make sense for either scenario.: This statement is false as both scenarios can have positive domain values for time.
Conclusion
The most accurate interpretations based on the graphs are:
- Negative range values don't make sense for either scenario.
- The starting point of the balloon is higher than that of the elevator.
While both the first and third statements are correct, the emphasis on the starting height likely aligns best with the comparative aspect of the scenarios.
Final response: The starting point of the balloon is higher than that of the elevator.