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Which of the following formulas would find the surface area of a right cylinder where [tex]\(h\)[/tex] is the height, [tex]\(r\)[/tex] is the radius, and [tex]\(BA\)[/tex] is the base area?

Check all that apply.
A. [tex]\(BA + \pi r^2\)[/tex]
B. [tex]\(\pi r^2 + \pi rh\)[/tex]
C. [tex]\(2\pi r^2\)[/tex]
D. [tex]\(BA + 2\pi rh\)[/tex]
E. [tex]\(2\pi r^2 + 2\pi rh\)[/tex]

Sagot :

GinnyAnswer
To find the surface area of a right cylinder, we need to consider both the lateral surface area (the side of the cylinder) and the area of the two circular bases.

1. Base Area Calculation:
- The base area [tex]\(BA\)[/tex] of a cylinder is the area of one of its circular bases.
- The area of a circle is given by [tex]\(BA = \pi r^2\)[/tex].

2. Lateral Surface Area Calculation:
- The lateral surface area of a cylinder can be thought of as the area of a rectangle that has been wrapped around the cylinder.
- This rectangle has a height equal to the height [tex]\(h\)[/tex] of the cylinder and a width equal to the circumference of the base.
- The circumference of the base is [tex]\(2\pi r\)[/tex].
- So, the lateral surface area is given by [tex]\(L = 2\pi rh\)[/tex].

3. Total Surface Area Calculation:
- The total surface area [tex]\(SA\)[/tex] of the cylinder is the sum of the lateral surface area and the areas of the two bases.
- Since there are two bases, we need to multiply the base area by 2.
- Therefore, the total surface area is:
[tex]\[ SA = 2\pi r^2 + 2\pi rh \][/tex]

Given this information, we can match the provided options to the correct formulas for components of the surface area of a right cylinder:

A. [tex]\( BA + \pi r^2 \)[/tex] – Incorrect. [tex]\(BA\)[/tex] already represents the base area, which is [tex]\(\pi r^2\)[/tex]. This expression does not account for the lateral surface area.
B. [tex]\( \pi r^2 + \pi r h \)[/tex] – Incorrect. This formula represents one base area [tex]\(\pi r^2\)[/tex] and part of the lateral surface area [tex]\(\pi rh\)[/tex], but it does not include the total lateral surface area or both bases.
C. [tex]\( 2 \pi r^2 \)[/tex] – Incorrect. This only accounts for the area of the two bases, not the lateral surface.
D. [tex]\( BA + 2 \pi r h \)[/tex] – Incorrect. [tex]\(BA\)[/tex] already represents the base area, and this formula does not capture the area of both bases correctly.
E. [tex]\( 2 \pi r^2 + 2 \pi r h \)[/tex] – Correct. This formula accurately combines the areas of both bases and the lateral surface area.

Therefore, the correct formula for the surface area of a right cylinder is given by option E:

[tex]\[ 2 \pi r^2 + 2 \pi r h \][/tex]
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