To analyze the given quadratic equation \( y = -3x^2 + 48x \) which models the height of a hot air balloon after \( x \) minutes, we can identify key features:
- The general form of a quadratic equation is \( ax^2 + bx + c \). In this case, \( a = -3 \), \( b = 48 \), and \( c = 0 \).
- Since \( a \) is negative, the parabola opens downward, indicating that there will be a maximum height reached after a certain time.
- The maximum height can be found using the formula \( x = -\frac{b}{2a} \).
Calculating the time at which the maximum height occurs:
\[ x = -\frac{48}{2(-3)} = \frac{48}{6} = 8 \]
Thus, the balloon reaches its maximum height after \( 8 \) minutes.
To evaluate the provided statements:
- The balloon descends before it ascends. (Incorrect)
- The balloon starts at a height of 3 feet. (Incorrect; \( y \) starts at \( 0 \) feet at \( x = 0 \))
- The balloon reaches its maximum height after 8 minutes. (Correct; as calculated above)
- The balloon ascends for 16 minutes. (Incorrect; it actually ascends until it reaches the max height at 8 minutes, then descends)
Thus, the correct interpretation of a key feature of the model is:
The balloon reaches its maximum height after 8 minutes.
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