To analyze the two functions and the correct interpretation of their graphs, let's start by understanding the equations provided.
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Hot Air Balloon Path: \( f(x) = -x^2 + 150 \)
- This function represents a downward-opening parabola with its vertex (maximum height) at \( (0, 150) \). The balloon starts at a height of 150 meters when \( x = 0 \) (time = 0 minutes).
- As time (x) increases, the height (f(x)) will decrease and eventually reach the ground (height = 0).
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Mine Shaft Elevator Path: \( g(x) = -20x \)
- This function is a linear equation with a negative slope, indicating that for every minute that passes, the elevator descends 20 meters. When \( x = 0 \), the height is 0 meter.
Now, let's analyze the two scenarios.
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Starting Heights:
- The balloon starts at 150 meters (f(0) = 150).
- The elevator starts at 0 meters (g(0) = 0).
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When Do They Reach the Ground?
- For the balloon, we solve \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]
- For the elevator, we solve \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]
- The elevator reaches the ground instantly at 0 minutes, while the balloon takes approximately 12.25 minutes to land.
Based on these analyses, we can choose the correct interpretation of the graphs:
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The balloon will reach the ground before the elevator car does. - This is incorrect because the elevator reaches the ground at 0 minutes, while the balloon does not reach the ground until approximately 12.25 minutes.
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Positive domain values don't make sense for either scenario. - This is incorrect because both functions can be evaluated for positive time values (domain).
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The starting point of the balloon is higher than that of the elevator. - This is correct. The balloon starts at 150 meters, while the elevator starts at 0 meters.
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Negative range values don't make sense for either scenario. - This is correct in a practical context as heights below ground level do not make sense for these scenarios.
In conclusion, the most accurate statement interpreting the graphs is: "The starting point of the balloon is higher than that of the elevator."
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