The path of the diver is given by the quadratic equation:

$y = -\frac{4}{9}x^2 + \frac{24}{9}x + 10$

To find the maximum height of the diver, we need to determine the vertex of this parabola. Since this is a downward-opening parabola (the coefficient of $x^2$ is negative), the vertex represents the maximum point.

Formula for the $x$-coordinate of the vertex:

$x = -\frac{b}{2a}$ Here, $a = -\frac{4}{9}$ and $b = \frac{24}{9}$.

Step 1: Calculate $x$:

$x = -\frac{\frac{24}{9}}{2 \cdot \left(-\frac{4}{9}\right)} = -\frac{\frac{24}{9}}{-\frac{8}{9}} = 3$

Step 2: Find $y$ at $x = 3$:

Substitute $x = 3$ into the equation for $y$: $y = -\frac{4}{9}(3)^2 + \frac{24}{9}(3) + 10$ $y = -\frac{4}{9}(9) + \frac{24}{9}(3) + 10$ $y = -4 + 8 + 10 = 14$

Maximum Height:

The maximum height of the diver is 14 feet.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the significance of the vertex in a quadratic equation?
2. How can you determine if a parabola opens upwards or downwards?
3. How would you solve this problem graphically?
4. How do you rewrite the given equation in vertex form?
5. Can you think of a real-world scenario where this type of equation might also apply?

Tip: For any quadratic equation $y = ax^2 + bx + c$, the vertex is the key to finding the maximum or minimum value!