To interpret the graphs of the hot air balloon's path \( f(x) = -x^2 + 150 \) and the mine shaft elevator's path \( g(x) = -20x \), we can analyze their equations and their implications regarding height over time.
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Starting Points:
- The hot air balloon's equation, \( f(x) = -x^2 + 150 \), indicates that the balloon starts at a height of 150 meters when \( x = 0 \) (time = 0 minutes).
- The mine shaft elevator's equation, \( g(x) = -20x \), indicates that the elevator starts at a height of 0 meters when \( x = 0 \).
So, the balloon starts higher than the elevator.
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Behavior Over Time:
- The balloon's height decreases in a parabolic manner, reaching the ground when \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]
- The elevator's height decreases linearly and will reach the ground when \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]
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Domain Consideration:
- It’s clear that the domain (time in minutes) is positive for both situations. Negative time doesn’t make sense in this context because you cannot go back in time.
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Range Consideration:
- Negative range values represent heights below ground level for both situations, which do not make sense in these contexts.
Based on the analysis, we can conclude:
- The starting point of the balloon is higher than that of the elevator.
- The balloon will reach the ground before the elevator car does is incorrect since the elevator starts at the ground level.
Thus, the correct interpretation of the graphs is:
The starting point of the balloon is higher than that of the elevator.
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