Question

A volleyball player served by a 6-foot player at an initial upward velocity of 33 feet per second. It is modeled by the equation h = -16t^2 + 33t +6. h represents the height in feet and t represents the time in seconds. Define the domain of the ball when it reaches its maximum height

1. -1.03 seconds
2. 1.03 seconds
3. 1.22 seconds
4. 23.01 feet

Answer 1

To find the domain of the ball when it reaches its maximum height, we first need to determine the time at which the maximum height occurs. The height of the volleyball, \( h(t) \), is given by the quadratic equation:

\[
h(t) = -16t^2 + 33t + 6
\]

In a quadratic equation of the form \( ax^2 + bx + c \), the time at which the maximum height occurs can be calculated using the vertex formula \( t = -\frac{b}{2a} \).

In this equation:
- \( a = -16 \)
- \( b = 33 \)

Now, substituting the values into the vertex formula:

\[
t = -\frac{33}{2 \times -16} = \frac{33}{32} \approx 1.03125 \text{ seconds}
\]

This tells us that the maximum height is reached at approximately 1.03 seconds.

Now, considering the heights (\( h \)) the ball will reach, we can also find the maximum height by substituting \( t \) back into the height equation \( h(t) \):

\[
h\left(\frac{33}{32}\right) = -16\left(\frac{33}{32}\right)^2 + 33\left(\frac{33}{32}\right) + 6
\]

Calculating these values can confirm the maximum height, but the question strictly asks for the domain when the ball reaches its maximum height, which is simply the time at which it does so.

Thus, the options provided appear to relate to potential answers, but the key detail we need regarding the domain at the maximum height is simply:

**The time at which the maximum height occurs is approximately 1.03 seconds.**

The answer from the provided options is:

**2. 1.03 seconds.**