A hot air balloon's path is modeled using the function f(x)=−x2+150

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 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.

GPT-4o mini · Answer

To interpret the graphs of the hot air balloon and the mine shaft elevator, let's analyze the functions given:

Hot Air Balloon Function: $f (x) = - x^{2} + 150$
- This is a downward-opening parabola, which means it will reach a maximum height at $x = 0$ . At $x = 0$ , the height $f (0) = 150$ meters, which is the starting height of the balloon. As time increases from this point, the height will decrease and eventually reach the ground (height $= 0$ ) at a certain time.
Mine Shaft Elevator Function: $g (x) = - 20 x$
- This is a linear function with a slope of -20, meaning it decreases at a constant rate. The elevator starts at height $g (0) = 0$ meters (ground level) and goes downwards as time increases.

Now, let’s evaluate the statements given:

Positive domain values don't make sense for either scenario.
This statement is not correct, as positive domain values (time) are meaningful in both scenarios.
The balloon will reach the ground before the elevator car does.
This statement needs calculation. The balloon reaches the ground when $f (x) = 0$ : $- x^{2} + 150 = 0 ⟹ x^{2} = 150 ⟹ x = \sqrt{150} \approx 12.247$ The elevator reaches the ground when $g (x) = 0$ : $- 20 x = 0 ⟹ x = 0$ Therefore, the elevator starts at ground level (0) while the balloon takes approximately 12.25 minutes to reach the ground. Thus, this statement is true.
The starting point of the balloon is higher than that of the elevator.
This statement is also true; the balloon starts at 150 meters while the elevator starts at 0 meters.
Negative range values don't make sense for either scenario.
This statement is correct in that negative heights do not make sense in the context of the scenarios being considered.

Conclusion:

Given these analyses, the most accurate statement regarding the comparison of the graphs and the situation is:

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

However, since the second statement about the balloon reaching the ground before the elevator is indeed true in terms of functionality, the correct interpretation should consider both aspects. Given the context often leads towards operational timings, the more significant takeaway would be the balloon's descent duration. Thus:

The conclusion would rest on the timing of the ballon's descent vs. the elevator's states at ground level.