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The problem states that *y* varies inversely with the cube of *x*, and we're given that *y* = 108 when *x* = 2. We need to find the value of *y* when *x* = 6.

### Step 1: Set up the relationship.
Since *y* varies inversely with the cube of *x*, we can express this as:

\[
y = \frac{k}{x^3}
\]

where *k* is a constant.

### Step 2: Use the given values to find *k*.
We know that when *x* = 2, *y* = 108. Substituting these values into the equation:

\[
108 = \frac{k}{2^3}
\]

Simplifying:

\[
108 = \frac{k}{8}
\]

To solve for *k*, multiply both sides by 8:

\[
k = 108 \times 8 = 864
\]

### Step 3: Find *y* when *x* = 6.
Now that we know *k* = 864, we can substitute *x* = 6 into the equation:

\[
y = \frac{864}{6^3}
\]

Simplifying further:

\[
y = \frac{864}{216} = 4
\]

So, the value of *y* when *x* = 6 is **4**.

Would you like further details or have any other questions?

### Related questions:
1. How does the inverse variation differ from direct variation?
2. What would happen if *x* increased to 8 in this same relationship?
3. Can you explain how the cube of *x* affects the rate at which *y* changes?
4. How would the relationship change if *y* varied inversely with the square of *x*?
5. What are some real-world applications of inverse variation with powers?

### Tip:
When solving inverse variation problems, always isolate the constant *k* first to simplify finding the value of the dependent variable for any new input.

y varies inversely with the cube of x. If y is 108 when x is 2, what is the value of y when x is 6?

Solve an inverse variation problem where y varies with the cube of x. Given y = 108 when x = 2, learn how to find y when x = 6. Explore key algebraic concepts and understand the inverse relationship between variables in this Grade 9-12 math problem.

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