To interpret the graphs of the hot air balloon and the mine shaft elevator, we need to consider the equations provided and their implications.
The hot air balloon's height is modeled by the function \( f(x) = -x^2 + 150 \). This represents a downward-opening parabola with a maximum height of 150 meters (at \( x = 0 \)). As time increases, the height of the balloon will decrease, reaching the ground when \( f(x) = 0 \).
The mine shaft elevator's height is modeled by the function \( g(x) = -20x \). This represents a linear decrease in height, starting at a height of 0 meters at \( x = 0 \) and moving downwards at a rate of 20 meters per minute.
Let's analyze the responses provided:
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Positive domain values don't make sense for either scenario.
- This is not correct since both scenarios involve time that can be positive.
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The balloon will reach the ground before the elevator car does.
- This can be checked by finding when \( f(x) = 0 \) and \( g(x) = 0 \):
- For \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]
- For \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \]
- The hot air balloon does not reach the ground until about 12.25 minutes, whereas the elevator starts at ground level. So, this statement is also incorrect.
- This can be checked by finding when \( f(x) = 0 \) and \( g(x) = 0 \):
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The starting point of the balloon is higher than that of the elevator.
- The starting point of the balloon is 150 meters (at \( x = 0 \)), whereas the starting point of the elevator is 0 meters. Therefore, this statement is correct.
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Negative range values don't make sense for either scenario.
- While it is true that in real scenarios, negative height values do not make sense, the hot air balloon's path may reach negative values theoretically based on the quadratic model as it decreases. This could be misleading in interpreting the context.
Based on these analyses, the correct interpretation is:
The starting point of the balloon is higher than that of the elevator.