Get expert advice and community support on IDNLearn.com. Our Q&A platform offers reliable and thorough answers to ensure you have the information you need to succeed in any situation.
Sagot :
To find the surface area of a right cylinder, we'll break down the surface area into its components:
1. Understanding the components of the surface area:
- Lateral Surface Area: The surface area of the side of the cylinder, which is the height multiplied by the circumference of the base. This can be given by the formula [tex]\(2 \pi r h\)[/tex].
- Area of the Bases: The area of the two circular bases of the cylinder. Each base has an area of [tex]\(\pi r^2\)[/tex]. Since there are two bases, their total area is [tex]\(2 \pi r^2\)[/tex].
2. Combining these components:
- The total surface area [tex]\(A\)[/tex] of the cylinder is the sum of the lateral surface area and the area of the two bases.
- Mathematically,
[tex]\[ A = \text{(Lateral Surface Area)} + \text{(Area of the two bases)} \][/tex]
[tex]\[ A = 2 \pi r h + 2 \pi r^2 \][/tex]
Now, let's evaluate the given options based on the understanding of the components:
- Option A: [tex]\(2 \pi r^2\)[/tex]\
This formula only represents the area of the two bases and does not include the lateral surface area. Thus, it is not the total surface area.
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]\
Here, [tex]\(BA\)[/tex] stands for the base area, which is [tex]\(\pi r^2\)[/tex]. If we replace [tex]\(BA\)[/tex] with [tex]\(\pi r^2\)[/tex], the formula becomes:
[tex]\[ \pi r^2 + 2 \pi r h \][/tex]
This includes both the base area and the lateral surface area, so this formula is valid for finding the total surface area.
- Option C: [tex]\(BA + \pi r^2\)[/tex]\
Similar to the previous option, if [tex]\(BA\)[/tex] is replaced by [tex]\(\pi r^2\)[/tex], this formula becomes:
[tex]\[ \pi r^2 + \pi r^2 = 2 \pi r^2 \][/tex]
This formula results in only the total area of the two bases without including the lateral surface area. Thus, it is not the total surface area.
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]\
This includes the area of the two bases ([tex]\(2 \pi r^2\)[/tex]) and the lateral surface area ([tex]\(2 \pi r h\)[/tex]). This is the exact formula for the total surface area of the cylinder, so this formula is valid.
- Option E: [tex]\(\pi r^2 + \pi r h\)[/tex]\
This formula combines the area of one base ([tex]\(\pi r^2\)[/tex]) with half of the lateral surface area ([tex]\(\pi r h\)[/tex]). It is incomplete and does not represent the total surface area of the cylinder.
Given these evaluations, the formulas that correctly find the surface area of a right cylinder are:
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]
Hence, the correct options are B and D.
1. Understanding the components of the surface area:
- Lateral Surface Area: The surface area of the side of the cylinder, which is the height multiplied by the circumference of the base. This can be given by the formula [tex]\(2 \pi r h\)[/tex].
- Area of the Bases: The area of the two circular bases of the cylinder. Each base has an area of [tex]\(\pi r^2\)[/tex]. Since there are two bases, their total area is [tex]\(2 \pi r^2\)[/tex].
2. Combining these components:
- The total surface area [tex]\(A\)[/tex] of the cylinder is the sum of the lateral surface area and the area of the two bases.
- Mathematically,
[tex]\[ A = \text{(Lateral Surface Area)} + \text{(Area of the two bases)} \][/tex]
[tex]\[ A = 2 \pi r h + 2 \pi r^2 \][/tex]
Now, let's evaluate the given options based on the understanding of the components:
- Option A: [tex]\(2 \pi r^2\)[/tex]\
This formula only represents the area of the two bases and does not include the lateral surface area. Thus, it is not the total surface area.
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]\
Here, [tex]\(BA\)[/tex] stands for the base area, which is [tex]\(\pi r^2\)[/tex]. If we replace [tex]\(BA\)[/tex] with [tex]\(\pi r^2\)[/tex], the formula becomes:
[tex]\[ \pi r^2 + 2 \pi r h \][/tex]
This includes both the base area and the lateral surface area, so this formula is valid for finding the total surface area.
- Option C: [tex]\(BA + \pi r^2\)[/tex]\
Similar to the previous option, if [tex]\(BA\)[/tex] is replaced by [tex]\(\pi r^2\)[/tex], this formula becomes:
[tex]\[ \pi r^2 + \pi r^2 = 2 \pi r^2 \][/tex]
This formula results in only the total area of the two bases without including the lateral surface area. Thus, it is not the total surface area.
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]\
This includes the area of the two bases ([tex]\(2 \pi r^2\)[/tex]) and the lateral surface area ([tex]\(2 \pi r h\)[/tex]). This is the exact formula for the total surface area of the cylinder, so this formula is valid.
- Option E: [tex]\(\pi r^2 + \pi r h\)[/tex]\
This formula combines the area of one base ([tex]\(\pi r^2\)[/tex]) with half of the lateral surface area ([tex]\(\pi r h\)[/tex]). It is incomplete and does not represent the total surface area of the cylinder.
Given these evaluations, the formulas that correctly find the surface area of a right cylinder are:
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]
Hence, the correct options are B and D.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
