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)

Question

Responses

Positive domain values don't make sense for either scenario.

Negative range values don't make sense for either scenario.

The starting point of the balloon is higher than that of the elevator.

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

Answer 1

To interpret the graphs of the functions given for the hot air balloon and the mine shaft elevator, let's look at the equations:

1. Hot Air Balloon: \( f(x) = -x^2 + 150 \)

   • This is a downward-opening parabola. The vertex (which is the maximum height) occurs at \( x = 0 \), giving a maximum height of \( f(0) = 150 \) meters. As \( x \) increases, the height decreases, indicating the balloon rises to a maximum height and then descends.

2. Mine Shaft Elevator: \( g(x) = -20x \)

   • This is a linear function with a slope of -20, meaning the elevator descends steadily over time. At \( x = 0 \), the height is \( g(0) = 0 \) meters, indicating it starts at ground level, and as time increases, the height decreases.

Now, considering the response options:

1. Positive domain values don't make sense for either scenario.

   • This is not correct; positive domain values make sense as time progresses for both scenarios.

2. Negative range values don't make sense for either scenario.

   • This is true; both functions represent height, and height cannot be negative in practical situations.

3. The starting point of the balloon is higher than that of the elevator.

   • This is also true; the balloon starts at 150 meters, while the elevator starts at 0 meters.

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

   • To assess this, we can find where both functions equal zero to see when they reach the ground. For the balloon, solving \( -x^2 + 150 = 0 \) gives us \( x^2 = 150 \) or \( x = \sqrt{150} \) (approximately 12.25 minutes). For the elevator, solving \( -20x = 0 \) gives \( x = 0 \); the elevator starts at the ground level and moves downward.

Given this analysis, the most accurate statements regarding the graphs are:

• "The starting point of the balloon is higher than that of the elevator."
• "Negative range values don't make sense for either scenario."

Ultimately, because the prompt asks for a single interpretation, the best response would be that "The starting point of the balloon is higher than that of the elevator." This statement correctly reflects the initial heights of both scenarios.