Let's break down the problem step by step.

Step 1: Equation of the Parabola

The given parabola is $y = x(10 - x)$, which can be rewritten as: $y = 10x - x^2.$ This parabola opens downward, and its vertex occurs at $x = 5$ (the midpoint of $0$ and $10$, the x-intercepts). The endpoints of the base of the isosceles triangle are on this parabola.

Step 2: Identify the Base Endpoints

We are told the base of the isosceles triangle is parallel to the x-axis, and its length is 8 units. This means the x-coordinates of the endpoints of the base must be symmetric around the midpoint of the base.

Let the x-coordinates of the endpoints of the base be $x_1$ and $x_2$. Since the length of the base is 8, we know that: $|x_2 - x_1| = 8.$ For symmetry, let the midpoint be at $x = 5$, so: $x_1 = 5 - 4 = 1 \quad \text{and} \quad x_2 = 5 + 4 = 9.$

Step 3: Find the y-Coordinates of the Base Endpoints

We now substitute $x_1 = 1$ and $x_2 = 9$ into the equation of the parabola $y = 10x - x^2$ to find the y-coordinates of the endpoints:

For $x_1 = 1$: $y_1 = 10(1) - (1)^2 = 10 - 1 = 9.$

For $x_2 = 9$: $y_2 = 10(9) - (9)^2 = 90 - 81 = 9.$

So, the coordinates of the endpoints of the base are $(1, 9)$ and $(9, 9)$, and the base lies at $y = 9$.

Step 4: Find the Area of the Triangle

The vertex of the triangle lies on the x-axis, so the vertex has coordinates $(5, 0)$ because the vertex is directly below the midpoint of the base.

The area of a triangle is given by the formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.$ Here, the base of the triangle is the distance between the points $(1, 9)$ and $(9, 9)$, which is 8 units. The height of the triangle is the vertical distance between the x-axis and the base, which is 9 units.

Thus, the area is: $\text{Area} = \frac{1}{2} \times 8 \times 9 = 36.$

Final Answer:

The area of the triangle is $\boxed{36}$.

Would you like further details or have any questions?

Related Questions:

1. How do you find the coordinates of the vertex of a parabola from its equation?
2. What is the formula for the area of a triangle in terms of its base and height?
3. How can you determine the symmetry of a parabola from its equation?
4. What is the significance of the midpoint when dealing with symmetric shapes like isosceles triangles?
5. How can the height of a triangle be related to the distance between points on a curve?

Tip: When working with geometric problems involving curves, always start by identifying key points (like vertex, base endpoints) and use symmetry to simplify calculations.