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The given problem involves the surface area \( S \) of a right circular cylinder, which is defined by the formula:

\[
S = 2\pi r^2 + 2\pi rh
\]

Where:
- \( r \) is the radius of the base of the cylinder.
- \( h \) is the height of the cylinder.
- The total surface area \( S \) is provided as 276 square inches.
- The height \( h \) is 5 inches.

We are asked to find the radius \( r \).

Let’s substitute the known values into the equation and solve for \( r \).

\[
276 = 2\pi r^2 + 2\pi r(5)
\]

Simplifying this equation:

\[
276 = 2\pi r^2 + 10\pi r
\]

Now I will solve this equation to find the value of \( r \).The solutions for the radius \( r \) are:

\[
r = -\frac{5}{2} + \frac{\sqrt{25\pi + 552}}{2\sqrt{\pi}}
\]
or
\[
r = -\frac{\sqrt{25\pi + 552}}{2\sqrt{\pi}} - \frac{5}{2}
\]

Since the radius cannot be negative, we discard the negative root. Let me now compute the positive root for the radius.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can substitute the values into the positive root formula and calculate the radius manually:

\[
r = -\frac{5}{2} + \frac{\sqrt{25\pi + 552}}{2\sqrt{\pi}}
\]

Would you like to proceed with any other questions or calculations? Here are five related questions you could explore:

1. How would the radius change if the surface area increased to 300 square inches?
2. What would the formula look like if the cylinder had no top (i.e., an open cylinder)?
3. How can you compute the volume of the cylinder once you know the radius?
4. What is the effect of increasing the height on the surface area while keeping the radius constant?
5. How can this surface area formula be applied to solve real-world problems involving cylindrical containers?

**Tip:** When solving for variables in an equation, always make sure the units are consistent throughout.

The surface area of a right circular cylinder is given by S = 2πr^2 + 2πrh, where r is the radius of the cylinder and h is its height. What is the radius (in inches) of a cylinder of surface area 276 in² and height 5 in?

Learn how to calculate the radius of a cylinder with a surface area of 276 in² and a height of 5 inches using the formula S = 2πr^2 + 2πrh. This problem involves applying algebra and geometry concepts and solving a quadratic equation. Suitable for students in Grades 9-12.

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