Question
Question
A hot air balloon's path 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. 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.
Negative range 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 starting point of the balloon is higher than that of the elevator.
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A hot air balloon's path 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. 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.
Negative range 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 starting point of the balloon is higher than that of the elevator.
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Answers
Answered by
GPT-4o mini
To interpret the graphs of the hot air balloon and the mine shaft elevator, let's analyze the equations given.
1. **Hot Air Balloon Path**: The function \( f(x) = -x^2 + 150 \) represents a downward-opening parabola, which starts at a height of 150 meters when \( x = 0 \) (the starting point) and eventually decreases to the ground (height of 0) as time goes on.
2. **Mine Shaft Elevator Path**: The function \( g(x) = -20x \) represents a linear decrease in height, starting from a height of 0 meters and decreasing at a rate of 20 meters for every minute that passes.
Now let's address the statements:
- **"The balloon will reach the ground before the elevator car does."**
- To find when the balloon reaches the ground, we set \( f(x) = 0 \):
\[
0 = -x^2 + 150 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes}
\]
- The elevator car reaches a height of 0 when:
\[
0 = -20x \implies x = 0 \text{ minutes (it starts from the ground)}
\]
- The balloon reaches the ground around 12.25 minutes, while the elevator is already on the ground at \( x = 0 \), making this statement **true**.
- **"Positive domain values don't make sense for either scenario."**
- This is false as both scenarios operate within a positive time frame (time in minutes).
- **"Negative range values don't make sense for either scenario."**
- This is true in the context that both scenarios describe heights above ground level; negative heights do not make sense.
- **"The starting point of the balloon is higher than that of the elevator."**
- True, as the balloon starts at 150 meters and the elevator starts at 0 meters.
Based on the options provided, the correct interpretation of the graphs is:
**The starting point of the balloon is higher than that of the elevator.**
However, considering the first statement about when each reaches the ground is also critical, if you had to focus on that aspect, the first statement would be accurate if you're interpreting the physical progression of their paths. The elevator starts at ground level, while the balloon starts significantly higher, therefore the correct final interpretation would be the one regarding the starting points.
1. **Hot Air Balloon Path**: The function \( f(x) = -x^2 + 150 \) represents a downward-opening parabola, which starts at a height of 150 meters when \( x = 0 \) (the starting point) and eventually decreases to the ground (height of 0) as time goes on.
2. **Mine Shaft Elevator Path**: The function \( g(x) = -20x \) represents a linear decrease in height, starting from a height of 0 meters and decreasing at a rate of 20 meters for every minute that passes.
Now let's address the statements:
- **"The balloon will reach the ground before the elevator car does."**
- To find when the balloon reaches the ground, we set \( f(x) = 0 \):
\[
0 = -x^2 + 150 \implies x^2 = 150 \implies x = \sqrt{150} \approx 12.25 \text{ minutes}
\]
- The elevator car reaches a height of 0 when:
\[
0 = -20x \implies x = 0 \text{ minutes (it starts from the ground)}
\]
- The balloon reaches the ground around 12.25 minutes, while the elevator is already on the ground at \( x = 0 \), making this statement **true**.
- **"Positive domain values don't make sense for either scenario."**
- This is false as both scenarios operate within a positive time frame (time in minutes).
- **"Negative range values don't make sense for either scenario."**
- This is true in the context that both scenarios describe heights above ground level; negative heights do not make sense.
- **"The starting point of the balloon is higher than that of the elevator."**
- True, as the balloon starts at 150 meters and the elevator starts at 0 meters.
Based on the options provided, the correct interpretation of the graphs is:
**The starting point of the balloon is higher than that of the elevator.**
However, considering the first statement about when each reaches the ground is also critical, if you had to focus on that aspect, the first statement would be accurate if you're interpreting the physical progression of their paths. The elevator starts at ground level, while the balloon starts significantly higher, therefore the correct final interpretation would be the one regarding the starting points.