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) Responses approximately 2 seconds approximately 2 seconds approximately 0.85 seconds approximately 0.85 seconds approximately 5 seconds approximately 5 seconds approximately 15 seconds
Answers
GPT-4o mini
To analyze the quadratic function representing the height of the volleyball over time, let us summarize the context. The volleyball is served with an initial vertical speed of 26 ft/s from a starting height of 4.5 ft.
In general, a quadratic function describing the height \( f(x) \) can be expressed in the form of:
\[ f(x) = ax^2 + bx + c \]
Where:
- \( a \) is a negative coefficient that determines the direction of the parabola (opening downwards in this case since the ball will eventually fall back down),
- \( b \) is the initial speed (here, 26 ft/s),
- \( c \) is the initial height (4.5 ft).
Given that the maximum height of the parabola occurs at the vertex of the parabola, we can find the time at which this occurs using the vertex formula for a parabola given by:
\[ x = -\frac{b}{2a} \]
While we do not have the specific coefficients \( a \) and \( b \) defined, we are aware that the maximum height occurs first and we can interpret based on the context of the problem.
Since the options given for the time it takes to reach maximum height are:
- approximately 2 seconds
- approximately 0.85 seconds
- approximately 5 seconds
- approximately 15 seconds
Given that a volleyball served at the stated initial conditions would likely reach its maximum height within a timeframe of just a few seconds (as compared to the much larger numbers such as 15 seconds), approximately 2 seconds seems to be a reasonable estimate for it to reach maximum height.
Therefore, the most accurate response is:
**Approximately 2 seconds.**
In general, a quadratic function describing the height \( f(x) \) can be expressed in the form of:
\[ f(x) = ax^2 + bx + c \]
Where:
- \( a \) is a negative coefficient that determines the direction of the parabola (opening downwards in this case since the ball will eventually fall back down),
- \( b \) is the initial speed (here, 26 ft/s),
- \( c \) is the initial height (4.5 ft).
Given that the maximum height of the parabola occurs at the vertex of the parabola, we can find the time at which this occurs using the vertex formula for a parabola given by:
\[ x = -\frac{b}{2a} \]
While we do not have the specific coefficients \( a \) and \( b \) defined, we are aware that the maximum height occurs first and we can interpret based on the context of the problem.
Since the options given for the time it takes to reach maximum height are:
- approximately 2 seconds
- approximately 0.85 seconds
- approximately 5 seconds
- approximately 15 seconds
Given that a volleyball served at the stated initial conditions would likely reach its maximum height within a timeframe of just a few seconds (as compared to the much larger numbers such as 15 seconds), approximately 2 seconds seems to be a reasonable estimate for it to reach maximum height.
Therefore, the most accurate response is:
**Approximately 2 seconds.**