To analyze the graphs of the functions \( f(x) = -x^2 + 150 \) and \( g(x) = -20x \), let's interpret each function and its implications:
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Starting Points:
- The function \( f(x) \) models the path of the hot air balloon and represents a downward-opening parabola with a maximum height of 150 meters at \( x = 0 \). Hence, the hot air balloon starts at a height of 150 meters.
- The function \( g(x) \) models the mine shaft elevator, which decreases linearly starting from the origin (at time \( x = 0 \), the elevator’s height is 0 meters).
This leads us to conclude that the starting point of the balloon (150 meters) is higher than that of the elevator (0 meters).
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Negative Range Values:
- The height of both the balloon and the elevator can potentially reach negative values. For \( f(x) \), the range will eventually fall below 0 as the balloon descends. For \( g(x) \), the elevator's height decreases linearly, which also means it can become negative.
- Thus, negative height values do not make sense for either situation, as both the balloon and the elevator should remain above ground (height = 0).
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Ground Level:
- The balloon modeled by \( f(x) \) will hit the ground when \( f(x) = 0\). Solving \( -x^2 + 150 = 0 \) gives \( x^2 = 150 \) which means the balloon will reach the ground at approximately \( x = 12.25 \) minutes.
- The elevator \( g(x) \) will hit the ground (height = 0) at \( g(x) = 0\), which gives \( -20x = 0 \) indicating that the elevator is at ground level starting from \( x = 0 \).
- Therefore, the balloon will take about 12.25 minutes to reach the ground, which is longer than the elevator since it starts at the ground level.
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Positive Domain Values:
- Positive time values (domain values) are sensible in both scenarios, as time cannot be negative.
Conclusion: Based on the analysis, the statement "The starting point of the balloon is higher than that of the elevator" is accurate.
Thus, the correct interpretation is:
The starting point of the balloon is higher than that of the elevator.
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