Certainly! The Bisection Method is a numerical technique used to find roots of a continuous function by repeatedly narrowing down an interval that contains the root. Let's apply it to solve the cubic equation:

\[
h^3 - 3h^2 + 0.8 = 0
\]

### Step 1: Define the Function

First, let's define the function \( f(h) \):
\[
f(h) = h^3 - 3h^2 + 0.8
\]
We want to find the root of \( f(h) = 0 \), which gives us the value of \( h \).

### Step 2: Choose the Interval

We need to select an initial interval \([a, b]\) where the function changes sign, i.e., \( f(a) \) and \( f(b) \) have opposite signs. This guarantees that a root exists within the interval by the Intermediate Value Theorem.

Given the nature of the cubic function, let's start by checking values around the likely root based on the physical context (since \( h \) represents a height, \( h \) must be between 0 and the radius of the sphere, i.e., 0 to 1 meter).

### Step 3: Implement the Bisection Method

We will use the Bisection Method to approximate the root. I'll execute this using Python code to find the value of \( h \).The Bisection Method estimates the root of the equation \( h^3 - 3h^2 + 0.8 = 0 \) to be approximately \( h \approx 0.57428 \) meters. The value of the function at this root is very close to zero (\( 1.748 \times 10^{-6} \)), indicating a good approximation.

This means that the height of the portion of the sphere that is above the water is approximately **0.574 meters**.

Would you like to dive deeper into this result or explore any related topics? Here are some related questions you might consider:

1. How does the Bisection Method compare to other numerical methods like Newton's method?
2. What would happen if the interval was chosen incorrectly?
3. How many iterations were needed to reach this tolerance?
4. Can we visualize the function and the root graphically?
5. How would changes in the densities of the sphere and water affect the result?

**Tip**: The Bisection Method is very reliable, but it requires a proper initial interval where the function changes sign. Make sure to analyze the function before choosing the interval.

Explore how the Bisection Method is applied to solve the cubic equation h^3 - 3h^2 + 0.8 = 0. Understand the process step-by-step, including selecting intervals and executing the method. Learn the approximated root value and its significance. Suitable for advanced high school students and beyond.

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