To interpret the functions and their graphs:
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Hot Air Balloon: \( f(x) = -x^2 + 150 \)
- This function represents a downward-opening parabola with a maximum height of 150 meters (when \( x = 0 \)). The vertex of this parabola, which is the maximum height, occurs at \( x = 0 \), and as time increases, the height decreases.
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Mine Shaft Elevator: \( g(x) = -20x \)
- This function represents a straight line with a constant negative slope of -20, meaning that for each minute, the elevator drops 20 meters starting from a height of 0 meters (when \( x = 0 \)). Thus, it can be interpreted that it descends continuously.
Now, examining the options:
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Negative range values don't make sense for either scenario: This option is true because height (range) cannot be negative in either scenario (the balloon can't go below ground level and the elevator cannot go below its starting point).
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Positive domain values don't make sense for either scenario: This is incorrect. Positive values for time (domain) do make sense as both the balloon and the elevator can be observed over minutes.
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The starting point of the balloon is higher than that of the elevator: This is also true since the balloon starts at a height of 150m, while the elevator starts at 0m.
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The balloon will reach the ground before the elevator car does: This is not necessarily true. The parabola and the linear function must be analyzed further for intersection, but generally, the balloon will reach the ground after descending for a while.
Based on this analysis, the best choice would be:
Negative range values don't make sense for either scenario.
This option directly addresses the validity of negative heights in both scenarios. Additionally, a secondary correct observation could be that the starting point of the balloon is higher than that of the elevator. But the most universally significant statement is about negative range values.