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?

Question

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.

Answer 1

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.