We are given the polynomial model:

$y = -0.0001(x^2 - 100x)$

where $0 \leq x \leq 100$, and $y$ is the height reached by the roller coaster.

Step 1: Expand the polynomial model

First, expand the expression inside the parentheses:

$y = -0.0001(x^2 - 100x) = -0.0001x^2 + 0.01x$

So, the model becomes:

$y = -0.0001x^2 + 0.01x$

Step 2: Find the maximum height

To find the highest point reached by the roller coaster, we need to find the maximum value of this quadratic function. Since the coefficient of $x^2$ is negative ($-0.0001$), the parabola opens downwards, meaning it has a maximum point.

To find the $x$-coordinate of the maximum, use the formula for the vertex of a parabola:

$x_{\text{max}} = \frac{-b}{2a}$

where $a = -0.0001$ and $b = 0.01$.

$x_{\text{max}} = \frac{-0.01}{2(-0.0001)} = \frac{-0.01}{-0.0002} = 50$

So, the maximum height is reached when $x = 50$.

Step 3: Calculate the maximum height

Substitute $x = 50$ into the equation to find the maximum height:

$y = -0.0001(50^2) + 0.01(50)$ $y = -0.0001(2500) + 0.01(50) = -0.25 + 0.5 = 0.25$

Thus, the highest point reached by the roller coaster is 0.25 units.

Would you like more details or have any other questions?

Here are some related questions:

1. How does the coefficient of $x^2$ affect the shape of a parabola?
2. Can this polynomial model be used for all roller coaster designs?
3. What would happen if the coefficient of $x^2$ was positive?
4. How do you determine the domain of a polynomial model like this?
5. What other types of models can be used for roller coaster height predictions?

Tip: The vertex formula for a parabola is a quick way to find the maximum or minimum value without fully solving the quadratic equation.