To solve this problem, we'll analyze the quadratic function provided, using the given initial velocity to answer each part. The quadratic function modeling the path of the ball is:

$h(x) = -\frac{44x^2}{v^2} + x + 6$

where:

• $h(x)$ is the height of the ball above the floor,
• $x$ is the forward distance of the ball in front of the foul line,
• $v$ is the initial velocity in feet per second.

Given:

• Initial velocity $v = 27$ feet per second.

Step-by-Step Analysis

(a) Substitute the initial velocity and simplify the equation.

Given $v = 27$, substitute it into the equation:

$h(x) = -\frac{44x^2}{27^2} + x + 6$

Simplify:

$h(x) = -\frac{44x^2}{729} + x + 6$

(b) Determine the maximum height of the ball and the distance from the foul line where this occurs.

The maximum height occurs at the vertex of the parabola. For a quadratic equation of the form $h(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by:

$x = -\frac{b}{2a}$

In our equation:

• $a = -\frac{44}{729}$
• $b = 1$

Substituting:

$x = -\frac{1}{2 \times -\frac{44}{729}} = \frac{729}{88}$

The maximum height can be found by substituting $x = \frac{729}{88}$ back into the equation for $h(x)$:

$h\left(\frac{729}{88}\right) = -\frac{44 \left(\frac{729}{88}\right)^2}{729} + \frac{729}{88} + 6$

(c) Find the height of the ball when it reaches a forward distance of 15 feet.

Substitute $x = 15$ into the equation:

$h(15) = -\frac{44(15)^2}{729} + 15 + 6$

Calculate the value:

$h(15) = -\frac{44 \times 225}{729} + 15 + 6 = -\frac{9900}{729} + 21$

(d) Find the distance from the foul line where the ball will hit the floor (i.e., when $h(x) = 0$).

Set $h(x) = 0$ and solve for $x$:

$0 = -\frac{44x^2}{729} + x + 6$

Multiply through by 729 to eliminate the fraction:

$0 = -44x^2 + 729x + 4374$

Use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where:

• $a = -44$
• $b = 729$
• $c = 4374$

(e) Interpret the results in the context of the problem.

The results should tell us:

• The maximum height of the ball and where it occurs.
• The height at a specific forward distance.
• The point where the ball will land on the floor, representing how far from the foul line it reaches the ground.

Would you like detailed calculations for any of these parts?