When we are trying to determine the surface area of a right cylinder, which is composed of two circular bases and one curved surface (or lateral area), we need to utilize formulas that account for both components. Let's examine each option:
1. Option A: [tex]\(\pi r^2 + \pi r h\)[/tex]
This formula appears to sum the area of one base ([tex]\(\pi r^2\)[/tex]) and part of the lateral surface area ([tex]\(\pi r h\)[/tex]), but it does not cover all areas correctly. Therefore, this formula is not correct for finding the entire surface area of a right cylinder.
2. Option B: [tex]\(B A + \pi r^2\)[/tex]
Here, [tex]\(B A\)[/tex] stands for the base area of a cylinder and [tex]\(\pi r^2\)[/tex] is the area of one additional base. However, this only accounts for one base and possibly some other area. It does not correctly describe the entire surface area. Thus, this option is incorrect.
3. Option C: [tex]\(B A + 2 \pi r h\)[/tex]
The term [tex]\(B A\)[/tex] represents the base area, and [tex]\(2 \pi r h\)[/tex] represents the surface area of the curved side of the cylinder (the lateral area). This combination provides the surface area but lacks the area of the second base. Therefore, this formula is partially correct, but it's not complete for the entire surface area of the right cylinder.
4. Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]
In this formula, [tex]\(2 \pi r^2\)[/tex] represents the area of both circular bases (since [tex]\(\pi r^2\)[/tex] is the area of one base, doubling it accounts for both bases), and [tex]\(2 \pi r h\)[/tex] gives the lateral surface area of the cylinder. This formula accurately represents the total surface area of a right cylinder, making it a valid formula.
5. Option E: [tex]\(2 \pi r^2\)[/tex]
This formula accounts only for the areas of the two bases but neglects the lateral surface area. Therefore, this formula is not correct for the entire surface area of the right cylinder.
Given the above considerations, only the following options are valid for finding the surface area of a right cylinder:
- C: [tex]\(B A + 2 \pi r h\)[/tex]
- D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]
Final valid formulas: C and D.
