A hot air balloon is modeled using the function f(x)= -x^2 + 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

Question

Which statement correctly interprets the graph
The starting point of the ballon is higher than that of the elevator
Positive domain values don’t make sense for either scenario
Negative range values don’t make sense for either scenario
The ballon will reach the ground before the elevator car does

Answer 1

To analyze the functions given for the hot air balloon and the mine shaft elevator, we have the following:

• Hot Air Balloon: \( f(x) = -x^2 + 150 \)
• Mine Shaft Elevator: \( g(x) = -20x \)

1. Starting Points:

• For the hot air balloon, at \( x = 0 \) (the starting point): \[ f(0) = -0^2 + 150 = 150 \text{ meters} \]
• For the elevator: \[ g(0) = -20(0) = 0 \text{ meters} \]

So, the starting height of the hot air balloon (150 meters) is higher than that of the elevator (0 meters).

2. Positive Domain Values:

Both functions are modeled for \( x \) in minutes, starting from \( x = 0 \), which is a positive domain value. Therefore, positive domain values make sense for both scenarios, as they represent time in minutes.

3. Negative Range Values:

For the hot air balloon, since \( f(x) = -x^2 + 150 \), the maximum height is 150 meters and it will eventually fall and reach ground level (0 meters). Therefore, negative height (range values) does not make sense in this context. The elevator, on the other hand, will have negative height values as it descends below the ground level. This implies that negative range values make sense for the elevator but not for the balloon.

4. When Do They Reach the Ground?

To find when each reaches the ground (height of 0):

• For the hot air balloon \( f(x) = 0 \): \[ -x^2 + 150 = 0 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes} \]

• For the elevator \( g(x) = 0 \): \[ -20x = 0 \implies x = 0 \text{ minutes} \]

Thus, the elevator is at ground level initially (0 minutes), and the balloon reaches the ground at approximately 12.25 minutes.

Conclusion

Given all the points analyzed:

1. The starting point of the balloon is higher than that of the elevator is true.
2. Positive domain values don’t make sense for either scenario is false.
3. Negative range values don’t make sense for either scenario is false (it does for the elevator).
4. The balloon will reach the ground before the elevator car does is false (the elevator is already at ground level).

Thus, the correct interpretation from the given statements is:

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