Question
A volleyball is served by a 6-foot player at an initial upward velocity of 33 feet per second. The situation is modeled by the equation h=−16t2+33t+6
h representing the height in feet and t representing the time in seconds. Using this equation, define the domain of the ball when it reaches its maximum height.(1 point)
Responses
−1.03 seconds
negative 1.03 seconds
1.03 seconds
1.03 seconds
1.22 seconds
1.22 seconds
23.01 feet
h representing the height in feet and t representing the time in seconds. Using this equation, define the domain of the ball when it reaches its maximum height.(1 point)
Responses
−1.03 seconds
negative 1.03 seconds
1.03 seconds
1.03 seconds
1.22 seconds
1.22 seconds
23.01 feet
Answers
GPT-4o mini
To find the domain of the volleyball's height when it reaches its maximum height, we first need to determine the time at which the maximum height occurs. The height equation given is
\[
h = -16t^2 + 33t + 6
\]
This is a quadratic equation of the form \( h(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 33 \), and \( c = 6 \).
The time \( t \) at which the maximum height occurs can be calculated using the formula
\[
t = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \) into the formula:
\[
t = -\frac{33}{2(-16)} = \frac{33}{32} \approx 1.03 \text{ seconds}
\]
This indicates that the maximum height of the volleyball is reached at around \( 1.03 \) seconds.
Now, looking at the domain, we want to determine the interval of time from when the ball is served (which is \( t = 0 \)) until it reaches the maximum height.
Since the situation is asking for the interval in time just before it reaches the maximum height, the domain in this context extends from \( t = 0 \) to \( t = 1.03 \) seconds when the maximum height is achieved.
Thus, the domain of the ball while it is reaching its maximum height is:
**0 ≤ t ≤ 1.03 seconds**
This means the specific time of 1.03 seconds indeed marks the maximum height reached by the ball. The provided response options seem to have an inconsistency or may be misinterpreted in the context of the question. However, the best choice approximating the time at maximum height is:
**1.03 seconds.**
\[
h = -16t^2 + 33t + 6
\]
This is a quadratic equation of the form \( h(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 33 \), and \( c = 6 \).
The time \( t \) at which the maximum height occurs can be calculated using the formula
\[
t = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \) into the formula:
\[
t = -\frac{33}{2(-16)} = \frac{33}{32} \approx 1.03 \text{ seconds}
\]
This indicates that the maximum height of the volleyball is reached at around \( 1.03 \) seconds.
Now, looking at the domain, we want to determine the interval of time from when the ball is served (which is \( t = 0 \)) until it reaches the maximum height.
Since the situation is asking for the interval in time just before it reaches the maximum height, the domain in this context extends from \( t = 0 \) to \( t = 1.03 \) seconds when the maximum height is achieved.
Thus, the domain of the ball while it is reaching its maximum height is:
**0 ≤ t ≤ 1.03 seconds**
This means the specific time of 1.03 seconds indeed marks the maximum height reached by the ball. The provided response options seem to have an inconsistency or may be misinterpreted in the context of the question. However, the best choice approximating the time at maximum height is:
**1.03 seconds.**