Given the function for the surface area of a cylinder, where the height $h = 6$ inches, we have:

$A(r) = 2\pi r^2 + 12\pi r$

Determining the Domain of $A(r)$

The domain of $A(r)$ refers to the set of values for the radius $r$ for which the function is defined.

1. Identify constraints on $r$:

   • The radius $r$ represents a physical length, so it must be a non-negative number.
   • The radius cannot be zero or negative since $r = 0$ would imply a non-existent cylinder.

2. Mathematical constraint:

   • Therefore, the value of $r$ must satisfy: $r > 0$

3. Final Domain:

   • The domain of $A(r)$ is: $(0, \infty)$

Conclusion

The function $A(r) = 2\pi r^2 + 12\pi r$ is defined for all positive values of $r$.

Would you like more details on this? Here are some related questions:

1. How does changing the height of the cylinder affect the function $A(r)$?
2. How would the surface area change if the radius was doubled?
3. What would be the domain if the function included a negative radius?
4. How can you find the maximum or minimum of the surface area function?
5. How would you find the radius that minimizes the surface area for a given volume?

Tip: When dealing with physical constraints in mathematical problems, always consider whether negative or zero values make physical sense!