To analyze the given functions for the hot air balloon \( f(x) = -x^2 + 150 \) and the mine shaft elevator \( g(x) = -20x \), we can interpret their graphs.
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Starting Points:
- The hot air balloon starts at a height of 150 meters when \( x = 0 \) (since \( f(0) = 150 \)).
- The mine shaft elevator starts at a height of 0 meters when \( x = 0 \) (since \( g(0) = 0 \)).
Thus, the statement "The starting point of the balloon is higher than that of the elevator" is correct.
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Reaching the Ground:
- To find out when the hot air balloon reaches the ground, set \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]
- For the elevator, set \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \] This shows that the elevator reaches the ground as soon as it starts (at \( x = 0 \)), while the balloon takes approximately 12.25 minutes to reach the ground. Thus, the statement "The balloon will reach the ground before the elevator car does" is false.
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Positive Domain Values:
- In this context, both scenarios represent real-world situations where \( x \) (time) is positive and valid, so the statement "Positive domain values don't make sense for either scenario" is false.
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Negative Range Values:
- Since both the balloon and elevator are grounded at height \( 0 \) (and cannot physically go below ground level), negative heights do not make sense for either scenario. Thus, the statement "Negative range values don't make sense for either scenario" is true.
In conclusion, the most accurate interpretation from the provided options is:
The starting point of the balloon is higher than that of the elevator.