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It looks like the image contains a math problem related to the surface area and radius of a cylinder. Let's break it down step by step.

### Part a: 
You are given that the height of the cylinder is 8 inches and the surface area \( A(r) \) is expressed as:
\[
A(r) = 2\pi r^2 + 16\pi r
\]
You are asked to find the domain of \( A(r) \), or in other words, determine for which values of \( r \) the function \( A(r) \) is defined.

#### Analysis:
- The term \( r^2 \) implies that \( r \) is a radius, which cannot be negative.
- The term \( r \) also represents the radius, so it must be positive.

Thus, the function \( A(r) \) is defined for \( r \geq 0 \). In practical terms, the radius of a cylinder cannot be negative, so the domain of \( A(r) \) is:
\[
r \geq 0
\]

### Part b:
Now, you're asked to find the inverse function \( r(A) \), which means expressing \( r \) as a function of \( A \). We need to solve the given equation \( A(r) = 2\pi r^2 + 16\pi r \) for \( r \) in terms of \( A \).

#### Step-by-step Solution:
1. Start with the equation:
\[
A = 2\pi r^2 + 16\pi r
\]
2. Divide both sides of the equation by \( 2\pi \) to simplify:
\[
\frac{A}{2\pi} = r^2 + 8r
\]
3. Rearrange the terms to form a quadratic equation:
\[
r^2 + 8r - \frac{A}{2\pi} = 0
\]
4. Solve for \( r \) using the quadratic formula. The quadratic formula is:
\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( r^2 + 8r - \frac{A}{2\pi} = 0 \), the coefficients are \( a = 1 \), \( b = 8 \), and \( c = -\frac{A}{2\pi} \). Plugging these values into the quadratic formula gives:
\[
r = \frac{-8 \pm \sqrt{8^2 - 4(1)\left(-\frac{A}{2\pi}\right)}}{2(1)}
\]
5. Simplify:
\[
r = \frac{-8 \pm \sqrt{64 + \frac{2A}{\pi}}}{2}
\]
\[
r = \frac{-8 \pm \sqrt{\frac{64\pi + 2A}{\pi}}}{2}
\]
6. Since the radius must be positive, we discard the negative root:
\[
r = \frac{-8 + \sqrt{\frac{64\pi + 2A}{\pi}}}{2}
\]

This is the inverse function \( r(A) \), which gives the radius in terms of the surface area.

### For the specific value of \( A = 165.04 \):
You are provided with \( r(A) = 165.04 \). This is a specific value for \( A \), and if we plug this into the formula, we could compute the corresponding radius. However, from the image, it seems the quadratic equation may already be solved. 

Would you like me to help further in calculating that specific radius or clarify any steps in this solution?

#### Related Questions:
1. How does the quadratic formula help in solving for \( r \) from \( A(r) \)?
2. What happens if we use the negative root in the quadratic formula?
3. What other methods can be used to solve quadratic equations?
4. Why does the domain of \( A(r) \) include \( r \geq 0 \)?
5. How does the surface area formula change if the height of the cylinder changes?

**Tip:** Always check if the context (e.g., physical constraints like radius) leads to discarding certain solutions from the quadratic formula.

Assume that the height of your cylinder is 8 inches. Consider A as a function of r, so we can write that as A(r) = 2πr² + 16πr. What is the domain of A(r)? Write the radius r as a function of A. This is the inverse function to A(r), i.e., to turn A as a function of r into r as a function of A.

Learn how to analyze the surface area of a cylinder, determine the domain of A(r), and solve for the radius r as a function of A. This problem includes geometry concepts, quadratic equations, and inverse functions, suitable for high school students (Grades 9-12).

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