To analyze the functions and the situations they describe, let's take a closer look at both the hot air balloon's path and the mine shaft elevator's path.
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Hot Air Balloon:
- The function is given by \( f(x) = -x^2 + 150 \).
- This is a downward-opening parabola with a maximum height (the vertex) of 150 meters. The height starts at 150 meters when \( x = 0 \) (the starting point) and decreases as time (x) increases.
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Mine Shaft Elevator:
- The function is given by \( g(x) = -20x \).
- This is a linear function with a slope of -20, meaning the elevator car descends at a constant rate of 20 meters per minute. The height is 0 meters when \( x = 0 \) (starting point).
Graphing the functions:
- When graphed, \( f(x) \) will start at 150 meters and descend to the ground (0 meters) at a certain point, while \( g(x) \) will start at 0 meters and continue to decrease.
Analyzing the Statements:
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The balloon will reach the ground before the elevator car does.
- We need to find when each function hits the ground (0 meters).
- For \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]
- For \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]
- The elevator reaches the ground at \( x = 0 \) minutes, while the balloon reaches ground at approximately \( 12.25 \) minutes. This statement is true.
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Positive domain values don't make sense for either scenario.
- This is incorrect as both functions are meaningful in the positive domain (time).
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The starting point of the balloon is higher than that of the elevator.
- The balloon starts at 150 meters, while the elevator starts at 0 meters. This statement is true.
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Negative range values don't make sense for either scenario.
- Negative heights (ground level) are not feasible in this context; hence this statement is true.
Conclusion:
The interpretation that fits based on our analysis is:
- The balloon will reach the ground before the elevator car does.
So, the correct response is: The balloon will reach the ground before the elevator car does.