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=−16t^2+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.
A: -1.03 seconds
B: 23.01 feet
C: 1.03 seconds
D: 1.22 seconds
A: -1.03 seconds
B: 23.01 feet
C: 1.03 seconds
D: 1.22 seconds
Answers
GPT-4o mini
To find the time when the volleyball reaches its maximum height, we need to identify the vertex of the quadratic function \( h(t) = -16t^2 + 33t + 6 \).
The formula to find the time \( t \) at which the vertex (maximum) occurs for a quadratic equation of the form \( h(t) = at^2 + bt + c \) is given by:
\[
t = -\frac{b}{2a}
\]
Here, \( a = -16 \) and \( b = 33 \). Plugging in these values:
\[
t = -\frac{33}{2 \times -16} = \frac{33}{32}
\]
Calculating \( \frac{33}{32} \):
\[
t = 1.03125 \, \text{seconds} \, \approx 1.03 \, \text{seconds}
\]
Thus, the time at which the ball reaches its maximum height is approximately \( 1.03 \) seconds.
Regarding the domain of the ball when it reaches its maximum height, time cannot be negative, so the domain for this situation would typically be \( t \geq 0 \).
From the provided options, the closest value that represents the time when the ball reaches its maximum height is:
**C: 1.03 seconds**
The formula to find the time \( t \) at which the vertex (maximum) occurs for a quadratic equation of the form \( h(t) = at^2 + bt + c \) is given by:
\[
t = -\frac{b}{2a}
\]
Here, \( a = -16 \) and \( b = 33 \). Plugging in these values:
\[
t = -\frac{33}{2 \times -16} = \frac{33}{32}
\]
Calculating \( \frac{33}{32} \):
\[
t = 1.03125 \, \text{seconds} \, \approx 1.03 \, \text{seconds}
\]
Thus, the time at which the ball reaches its maximum height is approximately \( 1.03 \) seconds.
Regarding the domain of the ball when it reaches its maximum height, time cannot be negative, so the domain for this situation would typically be \( t \geq 0 \).
From the provided options, the closest value that represents the time when the ball reaches its maximum height is:
**C: 1.03 seconds**