Assum that the height of you cylinder is 6 inches. Consider A as a function of r, so we can write that A(r)=2pir^2+12pir. what is the domain of A(r)? In other words for which values of r is A(r) defined?

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!

Assume that the height of your cylinder is 6 inches. Consider A as a function of r, so we can write that A(r) = 2πr² + 12πr. What is the domain of A(r)? In other words, for which values of r is A(r) defined?

Understand how to determine the domain of the surface area function A(r) = 2πr² + 12πr for a cylinder with a height of 6 inches. Learn about the constraints on the radius and how it relates to the physical properties of the cylinder. This problem involves key geometry concepts, including the domain of a function and the surface area of cylinders, suitable for students in Grades 10-12.

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