The problem asks us to determine the volume $V$ of a cube when the surface area $A$ is $486 \, \text{in}^2$, using the given formula:

$A = 6V^{2/3}.$

Step-by-step solution:

1. Substitute $A = 486$ into the formula: $486 = 6V^{2/3}.$

2. Solve for $V^{2/3}$: Divide both sides of the equation by $6$: $V^{2/3} = \frac{486}{6} = 81.$

3. Solve for $V$: Raise both sides to the power of $3/2$ to eliminate the $2/3$: $V = 81^{3/2}.$

4. Simplify $81^{3/2}$:

   • The square root of $81$ is $9$.
   • $9^3 = 729$.

Thus, the volume of the cube is:

$\boxed{729 \, \text{in}^3}.$

Would you like more details or have further questions?
Here are 5 related questions to explore:

1. What is the relationship between the side length of a cube and its surface area?
2. How can you derive the formula $A = 6V^{2/3}$?
3. What are the steps to calculate the cube's side length using its volume?
4. Can this formula apply to non-cubic shapes? Why or why not?
5. How does scaling the side length of a cube affect its volume and surface area?

Tip: Always double-check units when working with surface area and volume problems!