Question
The height reached by a hot air balloon, in feet, after x minutes can be modeled by the equation y=-3x^2 +48x. Which of the following statements correctly interprets a key feature of the model?
1)The balloon reaches its maximum height after 8 minutes
2)the balloon starts at a height of 3 feet.
3)the balloon ascends for 16 minutes.
4)the balloon descends before it ascends.
1)The balloon reaches its maximum height after 8 minutes
2)the balloon starts at a height of 3 feet.
3)the balloon ascends for 16 minutes.
4)the balloon descends before it ascends.
Answers
Answered by
GPT-4o mini
To analyze the quadratic equation \( y = -3x^2 + 48x \), we can identify key features.
1. **Maximum Height**: The given quadratic opens downwards since the coefficient of the \( x^2 \) term is negative. The maximum height can be found using the vertex formula, \( x = -\frac{b}{2a} \), where \( a = -3 \) and \( b = 48 \).
\[
x = -\frac{48}{2 \cdot -3} = \frac{48}{6} = 8
\]
So, the balloon reaches its maximum height after 8 minutes, confirming that **statement 1** is correct.
2. **Height at 0 Minutes**: To determine if the balloon starts at a height of 3 feet, we need to evaluate \( y \) at \( x = 0 \):
\[
y = -3(0)^2 + 48(0) = 0
\]
Therefore, the balloon starts at a height of 0 feet, not 3, so **statement 2** is incorrect.
3. **Duration of Ascent and Descent**: The ascent lasts until the maximum height at \( x = 8 \) minutes, and then the balloon descends. The function confirms a single ascent followed by a descent, meaning the total ascent time is not 16 minutes. **Statement 3** is incorrect.
4. **Ascent and Descent Dynamics**: The balloon ascends until it reaches the peak height at 8 minutes and then starts descending afterward. So, it does not descend before it ascends. **Statement 4** is also incorrect.
Thus, the correct interpretation of a key feature is:
**1) The balloon reaches its maximum height after 8 minutes.**
1. **Maximum Height**: The given quadratic opens downwards since the coefficient of the \( x^2 \) term is negative. The maximum height can be found using the vertex formula, \( x = -\frac{b}{2a} \), where \( a = -3 \) and \( b = 48 \).
\[
x = -\frac{48}{2 \cdot -3} = \frac{48}{6} = 8
\]
So, the balloon reaches its maximum height after 8 minutes, confirming that **statement 1** is correct.
2. **Height at 0 Minutes**: To determine if the balloon starts at a height of 3 feet, we need to evaluate \( y \) at \( x = 0 \):
\[
y = -3(0)^2 + 48(0) = 0
\]
Therefore, the balloon starts at a height of 0 feet, not 3, so **statement 2** is incorrect.
3. **Duration of Ascent and Descent**: The ascent lasts until the maximum height at \( x = 8 \) minutes, and then the balloon descends. The function confirms a single ascent followed by a descent, meaning the total ascent time is not 16 minutes. **Statement 3** is incorrect.
4. **Ascent and Descent Dynamics**: The balloon ascends until it reaches the peak height at 8 minutes and then starts descending afterward. So, it does not descend before it ascends. **Statement 4** is also incorrect.
Thus, the correct interpretation of a key feature is:
**1) The balloon reaches its maximum height after 8 minutes.**