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)
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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

Answer 1

To find the vertex of the quadratic function \( h(t) = -16t^2 + 56t \), we can use the vertex formula for a parabola. The time \( t \) at which the maximum height occurs can be found using the formula:

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

For the function \( h(t) = -16t^2 + 56t \), we have:
- \( a = -16 \)
- \( b = 56 \)

Plugging in these values:

\[
t = -\frac{56}{2 \times -16} = -\frac{56}{-32} = 1.75 \text{ seconds}
\]

Now, to find the maximum height, we can substitute \( t = 1.75 \) back into the height equation:

\[
h(1.75) = -16(1.75)^2 + 56(1.75)
\]

Calculating \( h(1.75) \):

\[
h(1.75) = -16(3.0625) + 56(1.75)
\]
\[
h(1.75) = -49 + 98 = 49 \text{ feet}
\]

So, the ball took **1.75** seconds to reach its maximum height of **49** feet.

Putting these into the context of your statements:
- The vertex of the ball's path indicates that the ball took **1.75** seconds to reach its maximum height of **49** feet.
- The domain in context to this situation is \( 0 \) to \( 3.5 \) because the ball hits the ground after that time.

So now you have:
- Response areaseconds: **1.75**
- Response area feet: **49**
- The ball hits the ground after **3.5** seconds, completing its trajectory.