To analyze the functions given:
-
Hot Air Balloon Function: \( f(x) = -x^2 + 150 \)
- This is a downward-opening parabola with a maximum height of 150 meters when \( x = 0 \) (the starting point).
- As \( x \) increases, the height decreases, indicating that the balloon will eventually reach the ground.
-
Mine Shaft Elevator Function: \( g(x) = -20x \)
- This is a linear function with a starting height of 0 meters when \( x = 0 \). The elevator descends at a constant rate of 20 meters per minute.
Interpreting the Graphs:
-
The starting point of the balloon is higher than that of the elevator: This is true since the balloon starts at 150 meters while the elevator starts at 0 meters.
-
Negative range values don't make sense for either scenario: This is also true; both scenarios deal with physical heights and thus can't be negative.
-
Positive domain values don't make sense for either scenario: This does not hold true as both functions can be interpreted for positive time values.
-
The balloon will reach the ground before the elevator does: This statement can be verified by finding when each function equals zero (height = 0). For the balloon (\(f(x) = 0\)), solving \( -x^2 + 150 = 0 \) gives \( x^2 = 150 \), so \( x \approx 12.25 \) minutes. For the elevator (\(g(x) = 0\)), solving \( -20x = 0 \) gives \( x = 0 \) (the elevator starts at ground level).
The correct interpretation based on the evaluation of the functions is "The starting point of the balloon is higher than that of the elevator."