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

Question

. 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

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

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.

Answer 1

To analyze the functions and the situations they describe, let's take a closer look at both the hot air balloon's path and the mine shaft elevator's path.

1. Hot Air Balloon:

   • The function is given by \( f(x) = -x^2 + 150 \).
   • This is a downward-opening parabola with a maximum height (the vertex) of 150 meters. The height starts at 150 meters when \( x = 0 \) (the starting point) and decreases as time (x) increases.

2. Mine Shaft Elevator:

   • The function is given by \( g(x) = -20x \).
   • This is a linear function with a slope of -20, meaning the elevator car descends at a constant rate of 20 meters per minute. The height is 0 meters when \( x = 0 \) (starting point).

Graphing the functions:

• When graphed, \( f(x) \) will start at 150 meters and descend to the ground (0 meters) at a certain point, while \( g(x) \) will start at 0 meters and continue to decrease.

Analyzing the Statements:

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

   • We need to find when each function hits the ground (0 meters).
   • For \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]
   • For \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]
   • The elevator reaches the ground at \( x = 0 \) minutes, while the balloon reaches ground at approximately \( 12.25 \) minutes. This statement is true.

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

   • This is incorrect as both functions are meaningful in the positive domain (time).

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

   • The balloon starts at 150 meters, while the elevator starts at 0 meters. This statement is true.

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

   • Negative heights (ground level) are not feasible in this context; hence this statement is true.

Conclusion:

The interpretation that fits based on our analysis is:

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

So, the correct response is: The balloon will reach the ground before the elevator car does.