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)? In other words, for which values of r is  A(r) defined? If the surface area is  125 square inches, what is the radius r? In other words, evaluate r(125). Round your answer to 2 decimal places.

Let's break down the two parts of your question: determining the domain of the function \( A(r) \) and solving for \( r \) when the surface area is 125 square inches.

### Part 1: Domain of \( A(r) \)

The function you provided is:
\[
A(r) = 2\pi r^2 + 16\pi r
\]
This is the formula for the surface area of a cylinder, where:
- \( r \) is the radius of the base.
- The term \( 2\pi r^2 \) represents the area of the two circular bases.
- The term \( 16\pi r \) represents the lateral surface area, which depends on the height (given as 8 inches, hence the 16).

For the domain, we are interested in the values of \( r \) for which \( A(r) \) is defined.

- The radius \( r \) must be a non-negative real number, since a negative radius does not make sense geometrically.
- Therefore, the domain is \( r \geq 0 \).

So, the domain of \( A(r) \) is:
\[
r \in [0, \infty)
\]

### Part 2: Solving for \( r \) when \( A(r) = 125 \)

Now, we are given that the surface area is 125 square inches and need to find \( r \).

We have the equation:
\[
A(r) = 2\pi r^2 + 16\pi r = 125
\]
Dividing both sides by \( \pi \) to simplify:
\[
2r^2 + 16r = \frac{125}{\pi}
\]

Now, let's solve this quadratic equation for \( r \). I will calculate the value for \( r \).The solutions to the equation are approximately:

\[
r \approx -9.99 \quad \text{or} \quad r \approx 1.99
\]

Since the radius \( r \) must be a non-negative value, we discard the negative solution. Thus, the radius is approximately:

\[
r \approx 1.99 \, \text{inches}
\]

### Summary:
- The domain of \( A(r) \) is \( [0, \infty) \).
- The radius \( r \) for a surface area of 125 square inches is approximately \( 1.99 \) inches.

Do you want further details or have any questions?

Here are 5 relative questions to explore:
1. How is the surface area formula of a cylinder derived?
2. What happens to \( A(r) \) as \( r \) approaches infinity?
3. Can the formula \( A(r) \) be rearranged to solve for the height \( h \) instead of radius?
4. How would the surface area change if the height of the cylinder were different?
5. What is the significance of the different parts of the formula \( A(r) \)?

**Tip:** When solving quadratic equations, always consider the physical meaning of the variable to decide if both solutions are valid.

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)? In other words, for which values of r is A(r) defined? If the surface area is 125 square inches, what is the radius r? In other words, evaluate r(125). Round your answer to 2 decimal places.

Explore how to find the domain of the surface area function of a cylinder and solve for the radius when the surface area is 125 square inches. This problem covers algebra, quadratic equations, and geometric formulas. Suitable for students in Grades 9-12.

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