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.