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

Answer 1

To interpret the quadratic function model in the context of the volleyball being served, we can recognize that the typical form of a quadratic function representing projectile motion is:

\[ f(t) = -at^2 + bt + c \]

where:
- \( f(t) \) is the height of the projectile (the volleyball in this case),
- \( a \) is a positive constant that determines the rate of descent,
- \( b \) is the initial velocity (26 ft/sec in this case),
- \( c \) is the initial height (4.5 ft in this case).

The maximum height of a projectile modeled by a quadratic function occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

\[ t = -\frac{b}{2a} \]

In this case, the maximum height occurs at some \( t \), which we need to approximate based on the provided options. While we don't have the exact parameters \( a \) and \( b \) for this particular function, we can deduce based on the typical behavior of projectile motion.

With the initial upward velocity and some height, we can conclude that the time to reach maximum height would likely be in the range of less than 5 seconds. Since the option choices are:

1. approximately 2 seconds
2. approximately 0.85 seconds
3. approximately 5 seconds
4. approximately 15 seconds

Given that in most typical scenarios a volleyball would reach its peak height in a brief time span, the best choice from the provided options is **approximately 2 seconds**.

Therefore, the answer is:

**approximately 2 seconds**.