To interpret the graphs of the hot air balloon's path \( f(x) = -x^2 + 150 \) and the mine shaft elevator's path \( g(x) = -20x \):
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Hot Air Balloon: The function \( f(x) = -x^2 + 150 \) represents a parabola that opens downward, with a maximum height of 150 meters at \( x = 0 \) (starting point). As \( x \) (time in minutes) increases, the height will decrease until it reaches the ground (where \( f(x) = 0 \)). The balloon will touch the ground when \( -x^2 + 150 = 0 \), solving this yields \( x^2 = 150 \), leading to \( x \approx 12.25 \) minutes.
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Mine Shaft Elevator: The function \( g(x) = -20x \) is a linear function that decreases steadily, starting from a height of 0 meters at \( x = 0 \) and moving downward as time increases. The elevator reaches a height of 0 meters or ground level at any positive value for \( x \) (e.g., at \( x = 0 \)).
Now, let’s analyze the statements:
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Positive domain values don't make sense for either scenario. (Incorrect) – Both the balloon and the elevator operate in positive time values.
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Negative range values don't make sense for either scenario. (Correct) – Since both height functions must represent heights in meters, they cannot be negative.
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The balloon will reach the ground before the elevator car does. (Incorrect) – The balloon reaches the ground at around 12.25 minutes, while the elevator starts at the ground (at \( x = 0 \)).
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The starting point of the balloon is higher than that of the elevator. (Correct) – The balloon starts at 150 meters, while the elevator starts at 0 meters.
From the analysis, the correct interpretation focuses on the heights and their starting points. Therefore, both statements "Negative range values don't make sense for either scenario" and "The starting point of the balloon is higher than that of the elevator" have merit; however, if you are to choose one, you might choose the second statement as it provides specific information about the comparison of their starting heights.