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The image contains a math problem about analyzing the surface area of a round cylinder, specifically focusing on deriving an expression for the radius \( r \) in terms of the surface area \( A \).

Here is a step-by-step breakdown of the problem:

### Given:
- The **height** of the cylinder is fixed at **6 inches**.
- The surface area \( A \) of the cylinder is expressed as:
  \[
  A = 2\pi r^2 + 12\pi r
  \]
  where \( r \) is the radius of the base of the cylinder.

### Part a:
You're asked to determine the domain of the surface area function, \( A(r) = 2\pi r^2 + 12\pi r \), which describes how the surface area changes as a function of the radius.

#### **Domain of \( A(r) \):**
For a physical cylinder, the radius \( r \) must be non-negative. Therefore, the domain is:
\[
r \geq 0
\]
However, in practical terms, \( r > 0 \), since a radius of zero would imply no cylinder exists.

### Part b:
Here, you are asked to solve for the radius \( r \) as a function of \( A \), meaning you need to find the inverse of the function \( A(r) \).

The given equation is quadratic in terms of \( r \):
\[
A = 2\pi r^2 + 12\pi r
\]
To solve for \( r \), follow these steps:

1. Divide both sides by \( 2\pi \):
   \[
   \frac{A}{2\pi} = r^2 + 6r
   \]

2. Rearrange into standard quadratic form:
   \[
   r^2 + 6r - \frac{A}{2\pi} = 0
   \]

3. Solve using the quadratic formula:
   \[
   r = \frac{-6 \pm \sqrt{6^2 - 4(1)(-\frac{A}{2\pi})}}{2(1)}
   \]
   Simplifying further:
   \[
   r = \frac{-6 \pm \sqrt{36 + 2A/\pi}}{2}
   \]
   Taking only the positive root (since radius must be positive):
   \[
   r = \frac{-6 + \sqrt{36 + 2A/\pi}}{2}
   \]
   \[
   r(A) = -3 + \frac{\sqrt{36 + 2A/\pi}}{2}
   \]

### Part c:
This part involves finding the radius \( r \) when the surface area \( A \) is 175 square inches. Using the inverse function derived in part b:

1. Substitute \( A = 175 \) into the formula:
   \[
   r = \frac{-6 + \sqrt{36 + \frac{2(175)}{\pi}}}{2}
   \]
   We can now calculate this numerically.

Let me compute this for you.The radius \( r \) of the cylinder, when the surface area is 175 square inches, is approximately **3.07 inches**.

Would you like further details on any part of the solution or explanation?

Here are some related questions for practice:

1. What happens to the surface area as the radius increases?
2. How does changing the height affect the surface area formula?
3. Can you derive the volume of the cylinder in terms of \( r \)?
4. What would the domain of the inverse function \( r(A) \) be?
5. How would you solve this problem if the surface area were given in a different unit, such as square centimeters?

**Tip:** Always check the units when solving geometric problems to ensure the final answer is consistent with the given data.

We'll be analyzing the surface area of a round cylinder. Given that the height of the cylinder is 6 inches, what is the radius of the cylinder if the surface area is 175 square inches?

This problem involves finding the radius of a cylinder when the surface area is known. Learn how to use the surface area formula and solve a quadratic equation. Suitable for students in Grades 9-12.

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