To analyze the two functions, we first need to understand their domains and ranges:
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Hot air balloon path: The function \( f(x) = -x^2 + 150 \) is a downward-facing parabola with its vertex at \( (0, 150) \). This tells us that the balloon starts at a height of 150 meters when \( x = 0 \) (at time zero). As \( x \) increases, the height decreases until it reaches ground level (height = 0) at its two intersection points with the x-axis.
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Mine shaft elevator path: The function \( g(x) = -20x \) describes a straight line that starts at the origin (0,0) and descends linearly. This means that for every minute, the elevator descends 20 meters. Therefore, at \( x = 0 \), the elevator is at 0 meters and continues to descend as time increases.
Evaluating the statements:
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The balloon will reach the ground before the elevator car does.
- To find when the balloon reaches the ground, we set \( f(x) = 0 \): \[ -x^2 + 150 = 0 \ x^2 = 150 \ x = \sqrt{150} \approx 12.25 \text{ minutes} \]
- For the elevator to reach the ground, we set \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]
- Thus, the elevator reaches the ground at time 0 minutes, while the balloon reaches the ground at approximately 12.25 minutes. Therefore, this statement is true.
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The starting point of the balloon is higher than that of the elevator.
- The starting height of the balloon is 150 meters, while the elevator starts at 0 meters. Thus, the balloon indeed starts higher than the elevator. This statement is true.
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Negative range values don't make sense for either scenario.
- The balloon's function \( f(x) \) will yield negative values for large \( x \), which represent heights below ground level, and thus don't make sense in this context. Similarly, the elevator's path will also yield negative values for sufficiently large \( x \). This statement is true.
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Positive domain values don't make sense for either scenario.
- This statement is incorrect since both functions are modeled for positive values of \( x \) (time). Both scenarios (the balloon's flight and the elevator's descent) can be effectively analyzed for positive time values.
Conclusion:
All the statements (1, 2, and 3) can be true, but the statement that is most definitively correct and interpretable regarding the graphs based on their behaviors is statement 1: "The balloon will reach the ground before the elevator car does."