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The given equation represents the path of a projectile, and we are asked to determine its maximum height by finding the vertex of the parabola.

The equation of the parabola is given as:

\[
y = \frac{5}{81}(x + 1)(17 - x)
\]

### Step 1: Expand the quadratic expression
First, we expand the product \((x + 1)(17 - x)\):

\[
(x + 1)(17 - x) = 17 - x^2 + x
\]

So, the equation becomes:

\[
y = \frac{5}{81}(17 - x^2 + x)
\]

### Step 2: Distribute the \(\frac{5}{81}\)

\[
y = \frac{5}{81}(17) - \frac{5}{81}(x^2) + \frac{5}{81}(x)
\]

\[
y = \frac{85}{81} - \frac{5}{81}x^2 + \frac{5}{81}x
\]

Now, the equation is in standard quadratic form:

\[
y = -\frac{5}{81}x^2 + \frac{5}{81}x + \frac{85}{81}
\]

### Step 3: Find the x-coordinate of the vertex
For a quadratic equation \(y = ax^2 + bx + c\), the x-coordinate of the vertex is given by:

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

Here, \(a = -\frac{5}{81}\) and \(b = \frac{5}{81}\), so:

\[
x = -\frac{\frac{5}{81}}{2 \times -\frac{5}{81}} = \frac{1}{2}
\]

### Step 4: Find the y-coordinate of the vertex
To find the maximum height, we substitute \(x = \frac{1}{2}\) into the equation for \(y\):

\[
y = \frac{5}{81}(x + 1)(17 - x)
\]

Substitute \(x = \frac{1}{2}\):

\[
y = \frac{5}{81}\left(\frac{1}{2} + 1\right)\left(17 - \frac{1}{2}\right)
\]

\[
y = \frac{5}{81} \times \frac{3}{2} \times \frac{33}{2}
\]

\[
y = \frac{5}{81} \times \frac{99}{4} = \frac{495}{324} = \frac{55}{36}
\]

Thus, the maximum height is:

\[
y = \frac{55}{36} \approx 1.53 \, \text{units}.
\]

### Conclusion:
The maximum height is approximately \( \frac{55}{36} \) units.

Would you like further details or have any questions?

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Here are 5 relative questions to explore further:

1. How do we derive the vertex formula for a parabola?
2. What is the significance of the vertex in projectile motion?
3. How does changing the coefficient \(\frac{5}{81}\) affect the parabola's shape?
4. Can we find the total distance the object traveled horizontally?
5. How does the starting point \((-1,0)\) influence the equation?

**Tip:** In a quadratic equation, the vertex represents the maximum or minimum value, depending on the sign of the leading coefficient \(a\).

The path of an object hurled from the point (-1,0) is the parabola y = (5/81)(x + 1)(17 - x). Determine the maximum height by finding the vertex.

Discover how to calculate the maximum height of a parabolic trajectory in projectile motion using the vertex formula. This lesson covers quadratic expansions and applications of the vertex formula in finding maximum and minimum values of a function, tailored for high school students.

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