To analyze the graphs of the hot air balloon's path \( f(x) = -x^2 + 150 \) and the mine shaft elevator's path \( g(x) = -20x \), we can derive some key observations based on their equations and respective interpretations.
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Starting Points:
- The hot air balloon starts at a height of 150 meters when \( x = 0 \) (i.e., at \( (0, 150) \)).
- The mine shaft elevator starts at a height of 0 meters when \( x = 0 \) (i.e., at \( (0, 0) \)).
- Therefore, the statement "The starting point of the balloon is higher than that of the elevator" is true.
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Range Values:
- For the hot air balloon, the maximum height is 150 meters, but it will eventually yield negative heights due to the quadratic nature of the function \( f(x) = -x^2 + 150 \) when \( x \) exceeds 12 (as it reaches ground level at \( x \approx 12.25 \)).
- For the elevator, the height decreases linearly from 0 as time increases, so when \( x \) increases, negative heights will eventually occur.
- This implies that negative range values don't make sense for the physical interpretations of both scenarios; however, they exist mathematically in the equations.
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Time to Reach the Ground:
- The balloon reaches the ground when \( f(x) = 0 \) which occurs around \( x \approx 12.25 \) minutes.
- The elevator reaches the ground when \( g(x) = 0 \) which occurs at \( x = 0 \) (it does not go below ground level).
- Thus, the balloon will reach the ground after the elevator begins its descent.
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Domain Values:
- The domain refers to time in minutes. Since negative time doesn’t make sense within this context, positive domain values are indeed necessary for both scenarios to be valid.
Given these observations, the correct interpretation based on the options provided is:
The starting point of the balloon is higher than that of the elevator.