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The problem asks to find a formula for the function \( g(x, y, z) \) whose level surfaces are planes parallel to the plane \( z = 2x + 8y - 8 \).

### Explanation:
The equation of a plane is given by:
\[
z = 2x + 8y - 8
\]
This is the equation of one specific plane. The general form for a family of planes parallel to this would be:
\[
z = 2x + 8y + C
\]
where \( C \) is a constant that shifts the planes up or down, maintaining parallelism.

For the function \( g(x, y, z) \), we want to describe a set of level surfaces where each surface corresponds to a different value of \( C \). That can be done by rearranging the equation as:
\[
g(x, y, z) = z - (2x + 8y)
\]
This equation describes a level surface \( g(x, y, z) = C \), where the constant \( C \) represents different parallel planes.

### Final Answer:
The formula for the function is:
\[
g(x, y, z) = z - 2x - 8y
\]

Would you like further details or have any questions?

Here are some related questions for practice:
1. How do you generalize the concept of level surfaces for more complex surfaces?
2. How can you verify that two planes are parallel?
3. What role does the constant \( C \) play in level surfaces?
4. How do you find the normal vector to the plane given by \( z = 2x + 8y - 8 \)?
5. What are the implications of level surfaces in physical applications like gravitational or electric fields?

**Tip:** When dealing with planes, always check for their normal vectors to quickly identify if they are parallel.

Find a formula for a function g(x, y, z) whose level surfaces are planes parallel to the plane z = 2x + 8y - 8.

Find a function g(x, y, z) whose level surfaces are parallel to the plane z = 2x + 8y - 8. This problem involves understanding equations of planes and level surfaces, key concepts in multivariable calculus.

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