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Interpret the following quadratic function model and graph given the following context. A volleyball is served into the air at...Question
Interpret the following quadratic function model and graph given the following context. A volleyball is served into the air at 26 ft./ sec from a height of 4.5 ft. The quadratic equation represents the height of the ball over time and in seconds. The graph illustrates this path where x represents the time in seconds and f(x) represents the height in feet. Approximately how long does it take for the volleyball to reach maximum height?
(1 point)
A. approximately 2 seconds
B. approximately 5 seconds
C. approximately 15 seconds
D. approximately 0.85 seconds
(1 point)
A. approximately 2 seconds
B. approximately 5 seconds
C. approximately 15 seconds
D. approximately 0.85 seconds
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GPT-4o mini
To determine how long it takes for the volleyball to reach its maximum height, we first need to understand the general form of a quadratic function, which has the formula:
\[ f(x) = ax^2 + bx + c \]
In this context, \( f(x) \) represents the height of the volleyball in feet, and \( x \) represents time in seconds.
The maximum height of a parabola (whose graph represents the function) occurs at the vertex. For a quadratic equation in standard form, the x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
Without knowing the specific coefficients \( a \) and \( b \) from the given quadratic equation, we can still make a reasoned estimate based on the provided options.
In this scenario, since the volleyball is served with a vertical initial velocity and from an initial height, we can reasonably assume that the maximum height would occur in the first few seconds after the serve.
From the options provided:
A. approximately 2 seconds
B. approximately 5 seconds
C. approximately 15 seconds
D. approximately 0.85 seconds
Given that volleyball serves typically reach their maximum height relatively quickly, the most plausible options would be A (approximately 2 seconds) or D (approximately 0.85 seconds).
Since A is the nearest estimate for how long it would take for the volleyball to reach its maximum height, the best answer appears to be:
**A. approximately 2 seconds.**
This estimation aligns well with the typical physics of a volleyball serve, where the ball reaches high points in the range of 1 to 3 seconds.
\[ f(x) = ax^2 + bx + c \]
In this context, \( f(x) \) represents the height of the volleyball in feet, and \( x \) represents time in seconds.
The maximum height of a parabola (whose graph represents the function) occurs at the vertex. For a quadratic equation in standard form, the x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
Without knowing the specific coefficients \( a \) and \( b \) from the given quadratic equation, we can still make a reasoned estimate based on the provided options.
In this scenario, since the volleyball is served with a vertical initial velocity and from an initial height, we can reasonably assume that the maximum height would occur in the first few seconds after the serve.
From the options provided:
A. approximately 2 seconds
B. approximately 5 seconds
C. approximately 15 seconds
D. approximately 0.85 seconds
Given that volleyball serves typically reach their maximum height relatively quickly, the most plausible options would be A (approximately 2 seconds) or D (approximately 0.85 seconds).
Since A is the nearest estimate for how long it would take for the volleyball to reach its maximum height, the best answer appears to be:
**A. approximately 2 seconds.**
This estimation aligns well with the typical physics of a volleyball serve, where the ball reaches high points in the range of 1 to 3 seconds.
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