The problem states that $y$ is directly proportional to the cube of $(x - 2)$. This can be expressed mathematically as:

$y = k(x - 2)^3$

where $k$ is the constant of proportionality.

Step 1: Solve for $k$ using the given values

When $y = 12$ and $x = 4$:

$12 = k(4 - 2)^3$

Simplify:

$12 = k(2)^3$

$12 = k \cdot 8$

Solve for $k$:

$k = \frac{12}{8} = 1.5$

Step 2: Find $y$ when $x = 5$

Now, substitute $k = 1.5$ and $x = 5$ into the equation $y = k(x - 2)^3$:

$y = 1.5(5 - 2)^3$

Simplify:

$y = 1.5(3)^3$

$y = 1.5 \cdot 27$

$y = 40.5$

Final Answer:

When $x = 5$, $y = 40.5$.

Let me know if you'd like further clarification or a breakdown of the solution!

Related Questions:

1. How do you determine the constant of proportionality in a direct variation problem?
2. What is the relationship between direct and inverse proportionality?
3. Can you solve for $y$ if $x = 6$ using the same relationship?
4. How would the equation change if $y$ were inversely proportional to the cube of $(x - 2)$?
5. What happens to $y$ as $x$ increases in a direct cube relationship?

Tip: When solving proportionality problems, always isolate the constant $k$ first using known values.