Question

30 Points! A soccer ball hit from the ground into the air and is modeled by the function

Answer 1

Answer:

A)

At t = 3/2

B)

36 feet

Step-by-step explanation:

The soccer ball is modeled by the function:

$h(t)=-16t^2+48t$

Where h(t) represents the height in feet of the ball over time t.

Note that this is a quadratic equation.

Part A)

Since this is a quadratic, the maximum height will be reached when the ball reaches its vertex.

So, we will find the x-coordinate of the vertex.

The vertex is given by:

$\displaystyle \Big (-\frac{b}{2a} , h( -\frac{b}{2a} ) \Big)$

In this case, a = -16 and b = 48. Thus, the t-coordinate is:

$\displaystyle t=-\frac{48}{2(-16)}=-\frac{48}{-32}=\frac{3}{2}$

The ball will reach is maximum height at t = 3/2.

Part B)

To find the maximum height, we can simply substitute the value back into the function and evaluated. Therefore:

$\displaystyle\begin{aligned} h(\frac{3}{2})&=-16\Big(\frac{3}{2}\Big)^2+48\Big(\frac{3}{2}\Big)\\\\ &=-16\Big(\frac{9}{4}\Big)+24(3)\\ \\ &=-36+72\\\\ &= 36 \end{aligned}$

The maximum height of the ball is 36 feet.