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.

Would you like further details or have any questions?

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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).