A right cylinder has a surface area of 217 square inches and a length of 12 inches. Using the equation for the surface area of a right cylinder and the quadratic formula, solve for the radius of the rolling pin in inches correct to three significant figures. Then convert the value of the radius from inches into centimeters.

To solve for the radius of the right cylinder, we'll use the formula for the surface area of a cylinder. The formula for the surface area \( A \) of a right cylinder is given by:

\[
A = 2\pi r h + 2\pi r^2
\]

Where:
- \( A \) is the surface area (217 square inches),
- \( r \) is the radius,
- \( h \) is the height (length) of the cylinder (12 inches),
- \( \pi \) is approximately 3.14159.

We can rearrange the formula into a quadratic equation in terms of \( r \):

\[
217 = 2\pi r(12) + 2\pi r^2
\]

Simplify the equation:

\[
217 = 24\pi r + 2\pi r^2
\]

Dividing through by \( 2\pi \) to make the equation easier to work with:

\[
\frac{217}{2\pi} = 12r + r^2
\]

Now solve this equation step by step.

### Step 1: Simplify the left side

We can calculate \( \frac{217}{2\pi} \):

\[
\frac{217}{2\pi} \approx \frac{217}{6.28318} \approx 34.544
\]

So the equation becomes:

\[
34.544 = 12r + r^2
\]

### Step 2: Rearrange into standard quadratic form

Rearrange the equation:

\[
r^2 + 12r - 34.544 = 0
\]

### Step 3: Apply the quadratic formula

The quadratic formula is:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( r^2 + 12r - 34.544 = 0 \), the coefficients are:
- \( a = 1 \),
- \( b = 12 \),
- \( c = -34.544 \).

Now plug these values into the quadratic formula:

\[
r = \frac{-12 \pm \sqrt{12^2 - 4(1)(-34.544)}}{2(1)}
\]
\[
r = \frac{-12 \pm \sqrt{144 + 138.176}}{2}
\]
\[
r = \frac{-12 \pm \sqrt{282.176}}{2}
\]
\[
r = \frac{-12 \pm 16.799}{2}
\]

### Step 4: Solve for the two possible values of \( r \)

\[
r = \frac{-12 + 16.799}{2} \quad \text{or} \quad r = \frac{-12 - 16.799}{2}
\]

So:

\[
r = \frac{4.799}{2} \quad \text{or} \quad r = \frac{-28.799}{2}
\]
\[
r = 2.3995 \quad \text{or} \quad r = -14.399
\]

Since the radius cannot be negative, we discard \( r = -14.399 \). Therefore, the radius is approximately:

\[
r \approx 2.400 \, \text{inches}
\]

### Step 5: Convert inches to centimeters

To convert from inches to centimeters, we use the conversion factor \( 1 \, \text{inch} = 2.54 \, \text{cm} \):

\[
r \approx 2.400 \times 2.54 = 6.096 \, \text{cm}
\]

Thus, the radius of the cylinder is approximately:

\[
\boxed{2.400 \, \text{inches} \quad \text{or} \quad 6.096 \, \text{cm}}
\]

Would you like further details on any step?

### Related Questions:
1. How is the surface area of a right cylinder related to its radius and height?
2. How can we solve quadratic equations that arise from geometric problems?
3. What are the differences between a right cylinder and other types of cylinders?
4. How would the result change if the length of the cylinder was different?
5. Can the quadratic formula be used for all geometric shapes?

### Tip:
Always check that all terms in an equation have consistent units before solving for any unknowns!

Learn how to solve for the radius of a right cylinder given its surface area of 217 square inches and length of 12 inches. This problem involves using the surface area formula for a right cylinder and the quadratic formula to find the radius in inches, then converting it to centimeters. Suitable for students in Grades 9-12.

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