Bot GPT 4-o mini To analyze the functions and their graphs, let's first look at the equations:
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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 when \( x = 0 \) and will decrease as \( x \) increases until it reaches the ground.
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Mine Shaft Elevator: \( g(x) = -20x \)
- This is a linear function with a negative slope. It starts at the origin \( (0, 0) \) and decreases linearly, meaning the elevator starts at ground level.
Now let's interpret the starting points and behaviors of both functions:
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Starting Height: The hot air balloon starts at 150 meters, while the elevator starts at 0 meters. Therefore, the statement "The starting point of the balloon is higher than that of the elevator." is true.
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Time to Reach the Ground: The hot air balloon reaches the ground when \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x \approx 12.25 \text{ minutes} \] The elevator reaches the ground when \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \] Therefore, the elevator reaches the ground before the balloon does.
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Positive Domain Values: Both functions are modeled for positive values of time, which makes sense in this context.
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Negative Range Values: The hot air balloon will eventually reach a height of 0, but both the balloon’s and elevator’s paths do not extend to negative heights (in meters) in the given scenario, as they are limited to non-negative heights.
Considering all this information, the correct statement is:
"The starting point of the balloon is higher than that of the elevator."
This statement clearly reflects the initial conditions of both the hot air balloon and the mine shaft elevator.
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