Question
An open, downward facing parabola is graphed on a coordinate plane. The x-axis ranges from negative 2 to 5 in increments of 1. The y-axis ranges from negative 1 to 16 in increments of 2.
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 0.85 seconds
approximately 0.85 seconds
approximately 15 seconds
approximately 15 seconds
approximately 2 seconds
approximately 2 seconds
approximately 5 seconds
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 0.85 seconds
approximately 0.85 seconds
approximately 15 seconds
approximately 15 seconds
approximately 2 seconds
approximately 2 seconds
approximately 5 seconds
Answers
Answered by
GPT-4o mini
To determine the time it takes for the volleyball to reach its maximum height, we can use the properties of a quadratic function. A downward-facing parabola reaches its maximum at the vertex. For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this context, the problem states the volleyball is served with an initial vertical velocity and height, but we don't have the exact quadratic equation provided. However, typical volleyball trajectories have a peak that is reached shortly after being served.
Assuming a realistic scenario based on typical physics of a volleyball:
1. The initial height is 4.5 ft, and the initial upward velocity is 26 ft/sec.
2. The vertex of the function (maximum height) typically occurs within a couple of seconds after the serve.
Given the response options, it is reasonable based on the physics of such an object (factoring in gravity and the initial speed) that the maximum height would occur approximately between 0.85 seconds and 2 seconds.
Based on the options:
- **approximately 0.85 seconds**
- **approximately 15 seconds** (too long for a typical trajectory)
- **approximately 2 seconds**
- **approximately 5 seconds** (also likely too long)
The best estimate for the time it takes the volleyball to reach its maximum height is **approximately 0.85 seconds** or **approximately 2 seconds**, but **0.85 seconds** would be closer to the expected timing for maximum height from the initial conditions given.
Therefore, the answer is: **approximately 0.85 seconds.**
\[
x = -\frac{b}{2a}
\]
In this context, the problem states the volleyball is served with an initial vertical velocity and height, but we don't have the exact quadratic equation provided. However, typical volleyball trajectories have a peak that is reached shortly after being served.
Assuming a realistic scenario based on typical physics of a volleyball:
1. The initial height is 4.5 ft, and the initial upward velocity is 26 ft/sec.
2. The vertex of the function (maximum height) typically occurs within a couple of seconds after the serve.
Given the response options, it is reasonable based on the physics of such an object (factoring in gravity and the initial speed) that the maximum height would occur approximately between 0.85 seconds and 2 seconds.
Based on the options:
- **approximately 0.85 seconds**
- **approximately 15 seconds** (too long for a typical trajectory)
- **approximately 2 seconds**
- **approximately 5 seconds** (also likely too long)
The best estimate for the time it takes the volleyball to reach its maximum height is **approximately 0.85 seconds** or **approximately 2 seconds**, but **0.85 seconds** would be closer to the expected timing for maximum height from the initial conditions given.
Therefore, the answer is: **approximately 0.85 seconds.**
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