Assume that the height of your cylinder is 66 inches. Consider AA as a function of rr, so we can write that as A(r)=2πr2+12πrAr=2πr2+12πr. What is the domain of A(r)Ar? In other words, for which values of rr is A(r)Ar defined?

Given the function \( A(r) = 2\pi r^2 + 12\pi r \), we need to determine its domain — the values of \( r \) for which the function is defined.

### Step 1: Analyze the terms in the function
The function consists of two terms:
- \( 2\pi r^2 \) 
- \( 12\pi r \)

Both of these terms involve powers of \( r \), and neither term includes a denominator or a square root. Thus, there are no restrictions on the function from division by zero or taking the square root of a negative number.

### Step 2: Consider physical constraints
Since \( r \) represents the radius of a cylinder, the radius must be non-negative. Therefore, \( r \geq 0 \). However, a radius of zero would result in a degenerate cylinder (just a line), so we typically exclude \( r = 0 \) from the domain when dealing with real-world problems involving cylinders.

### Conclusion:
The domain of \( A(r) \) is \( r > 0 \), meaning that \( r \) can take any positive value.

### Summary:
The domain of \( A(r) = 2\pi r^2 + 12\pi r \) is \( r > 0 \), because \( r \) must be positive for a valid cylinder.

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**Relative questions:**
1. How would the domain change if the function included a square root term, such as \( A(r) = \sqrt{r} + 2\pi r^2 \)?
2. What would be the volume formula for a cylinder with height 66 inches?
3. How does the surface area of a cylinder depend on the radius and height?
4. How would the function \( A(r) \) change if the height of the cylinder was doubled?
5. Can you express the function \( A(r) \) in a factored form?

**Tip:** When determining the domain of a function, always check for values that could cause division by zero, taking the square root of a negative number, or physical constraints (such as non-negative dimensions in geometry).

Assume that the height of your cylinder is 66 inches. Consider AA as a function of rr, so we can write that as A(r) = 2πr^2 + 12πr. What is the domain of A(r)? In other words, for which values of r is A(r) defined?

Explore how to determine the domain of the function A(r) = 2πr^2 + 12πr, which represents the surface area of a cylinder with a height of 66 inches. Learn about algebraic and geometric concepts relevant to domain determination, suitable for high school students.

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