A hot air balloon's path is modeled using the function f(x)=−x2+150 . A mine shaft elevator's path is modeled using the equation g(x)=−20x . In both instances, the domain is time, in minutes, and the range is height, in meters. Using GeoGebra, graph the two situations. Which statement correctly interprets the graphs? (1 point) Responses Positive domain values don't make sense for either scenario. Positive domain values don't make sense for either scenario. The balloon will reach the ground before the elevator car does. The balloon will reach the ground before the elevator car does. The starting point of the balloon is higher than that of the elevator. The starting point of the balloon is higher than that of the elevator. Negative range values don't make sense for either scenario.

1 answer

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.

  1. 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.

  2. 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 \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]
  3. 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.
  4. 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.