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Multiple Function Types in Context Quick Check
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Question
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 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.
Negative range values don't make sense for either scenario.

The balloon will reach the ground before the elevator car does.

1 answer

To analyze the functions given:

  1. 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.
  2. 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."