To interpret the graphs of the hot air balloon modeled by \( f(x) = -x^2 + 150 \) and the mine shaft elevator modeled by \( g(x) = -20x \), let's analyze them based on their equations and the context provided:
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Hot Air Balloon \( f(x) = -x^2 + 150 \):
- This is a downward-opening parabola with a vertex at \( (0, 150) \). It starts at a height of 150 meters (maximum height) at time \( x = 0 \) and will decrease as time progresses. The balloon will reach the ground when \( f(x) = 0 \), yielding the equation \( -x^2 + 150 = 0 \), which solves to \( x = \sqrt{150} \approx 12.25 \) minutes.
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Mine Shaft Elevator \( g(x) = -20x \):
- This is a linear function with a slope of -20, indicating that the elevator descends at a constant rate. It will reach the ground (height = 0) at \( g(x) = 0 \), which occurs at \( x = 0 \) (starting height) and \( g(x) < 0 \) for any positive time \( x \).
Now, considering the statements:
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Positive domain values don't make sense for either scenario.
- This is not correct. Positive values of time (domain) do make sense for both situations as time progresses.
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The starting point of the balloon is higher than that of the elevator.
- This is correct. The balloon starts at 150 meters while the elevator starts at 0 meters.
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The balloon will reach the ground before the elevator car does.
- This is incorrect. The balloon reaches the ground at approximately 12.25 minutes, while the elevator reaches the ground at \( x = 0 \) (it is already at ground level).
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Negative range values don't make sense for either scenario.
- This statement is more applicable to the scenarios. The hot air balloon cannot go below ground level (it stops at height 0), and the elevator's model does produce negative values which don't have physical meaning in this context.
The most accurate interpretation of the graphs would be:
The starting point of the balloon is higher than that of the elevator.