We are given the height function for the projectile above the water as follows:

$h(x) = -\frac{32x^2}{55^2} + x + 210$

where:

• $x$ is the horizontal distance of the projectile from the face of the cliff,
• $h(x)$ is the height of the projectile above the water.

To find the horizontal distance $x$ at which the height is maximized, we need to find the vertex of this quadratic function. For a quadratic equation in the form:

$h(x) = ax^2 + bx + c$

the horizontal distance $x$ at which the height is maximum can be found using the formula for the vertex:

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

Step 1: Identify the coefficients

The given equation is:

$h(x) = -\frac{32}{55^2}x^2 + x + 210$

So the coefficients are:

• $a = -\frac{32}{55^2}$
• $b = 1$

Step 2: Apply the vertex formula

Now, let's plug in the values into the vertex formula:

$x = -\frac{1}{2 \times -\frac{32}{55^2}}$

We'll now compute this step by step.The horizontal distance at which the height of the projectile is maximized is approximately 47 feet (rounded to the nearest foot).

Would you like more details or have any further questions?

Here are 5 related questions for deeper understanding:

1. How do you determine whether the vertex of a quadratic function gives a maximum or a minimum value?
2. What is the maximum height reached by the projectile at this horizontal distance?
3. How does the muzzle velocity affect the trajectory of the projectile?
4. How would the height function change if the cliff were taller or shorter?
5. How can you derive the time of flight from the given height function?

Tip: The sign of the quadratic term $a$ in a parabola determines whether the function opens upward (minimum) or downward (maximum).