Question
A soccer ball was kicked from the ground with an initial velocity of 56 feet per second and followed the path modeled by the function: h(t)=−16t2+56t .(3 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. The vertex of the ball's path indicates that the ball took Response areaseconds to reach its maximum height of Response area feet. Looking at the graph of this function, we can see that the domain in context to this situation is 0 to 3.5 because the ball Response area.
Answers
GPT-4o mini
To find the vertex of the function \( h(t) = -16t^2 + 56t \), we can use the formula for the vertex of a parabola in the form \( at^2 + bt + c \), which is given by \( t = -\frac{b}{2a} \).
Here, \( a = -16 \) and \( b = 56 \):
\[
t = -\frac{56}{2(-16)} = \frac{56}{32} = 1.75
\]
The maximum height can be found by substituting \( t = 1.75 \) back into the height function \( h(t) \):
\[
h(1.75) = -16(1.75)^2 + 56(1.75)
\]
Calculating it:
\[
= -16(3.0625) + 56(1.75)
\]
\[
= -49 + 98 = 49
\]
So, the vertex indicates that the ball took **1.75 seconds** to reach its maximum height of **49 feet**.
Regarding the domain of the situation: Since the ball is kicked from the ground and follows a parabolic path, it will touch the ground again. You can find when the ball hits the ground by solving for \( h(t) = 0 \):
\[
-16t^2 + 56t = 0
\]
\[
t(-16t + 56) = 0
\]
The solutions from this are \( t = 0 \) (when the ball is kicked) and \( t = \frac{56}{16} = 3.5 \) seconds (when the ball lands back on the ground).
Therefore, we can summarize:
The vertex of the ball's path indicates that the ball took **1.75** seconds to reach its maximum height of **49** feet. Looking at the graph of this function, we can see that the domain in context to this situation is **0 to 3.5** because the ball **hits the ground**.
Here, \( a = -16 \) and \( b = 56 \):
\[
t = -\frac{56}{2(-16)} = \frac{56}{32} = 1.75
\]
The maximum height can be found by substituting \( t = 1.75 \) back into the height function \( h(t) \):
\[
h(1.75) = -16(1.75)^2 + 56(1.75)
\]
Calculating it:
\[
= -16(3.0625) + 56(1.75)
\]
\[
= -49 + 98 = 49
\]
So, the vertex indicates that the ball took **1.75 seconds** to reach its maximum height of **49 feet**.
Regarding the domain of the situation: Since the ball is kicked from the ground and follows a parabolic path, it will touch the ground again. You can find when the ball hits the ground by solving for \( h(t) = 0 \):
\[
-16t^2 + 56t = 0
\]
\[
t(-16t + 56) = 0
\]
The solutions from this are \( t = 0 \) (when the ball is kicked) and \( t = \frac{56}{16} = 3.5 \) seconds (when the ball lands back on the ground).
Therefore, we can summarize:
The vertex of the ball's path indicates that the ball took **1.75** seconds to reach its maximum height of **49** feet. Looking at the graph of this function, we can see that the domain in context to this situation is **0 to 3.5** because the ball **hits the ground**.