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Sagot :
To solve this problem, we need to find the total surface area of the cylinder, which includes the lateral area and the area of the two bases.
First, let’s recall the formulas for the lateral area and the base area of a cylinder:
1. Lateral Area:
The lateral surface area of a cylinder is given by the formula:
[tex]\[ \text{Lateral Area} = \text{Circumference of the base} \times \text{Height} \][/tex]
2. Base Area:
The area of one base of the cylinder is given by:
[tex]\[ \text{Base Area} = \pi \times (\text{radius})^2 \][/tex]
3. Total Surface Area:
The total surface area of the cylinder is the sum of the lateral area and the areas of the two bases:
[tex]\[ \text{Total Surface Area} = \text{Lateral Area} + 2 \times \text{Base Area} \][/tex]
Given:
- The circumference of the base [tex]\(\text{= } 6\pi \ \text{in.}\)[/tex]
- The height of the cylinder [tex]\(\text{= } 10 \ \text{in.}\)[/tex]
- The area of the base [tex]\(\text{= } 9\pi \ \text{in.}^2\)[/tex]
Let's find each individual part:
1. Lateral Area:
[tex]\[ \text{Lateral Area} = 6\pi \ \text{in.} \times 10 \ \text{in.} = 60\pi \ \text{in.}^2 \][/tex]
2. Base Area:
Given the base area is [tex]\(9\pi \ \text{in.}^2\)[/tex]
3. Total Surface Area:
[tex]\[ \text{Total Surface Area} = 60\pi \ \text{in.}^2 + 2 \times 9\pi \ \text{in.}^2 \][/tex]
[tex]\[ \text{Total Surface Area} = 60\pi \ \text{in.}^2 + 18\pi \ \text{in.}^2 \][/tex]
[tex]\[ \text{Total Surface Area} = 78\pi \ \text{in.}^2 \][/tex]
Now, let's match this to the given options:
1. [tex]\(6 \pi + 9 \pi(10) \ \text{in.}^2 \)[/tex]
2. [tex]\(9 \pi + 6 \pi(10) \ \text{in.}^2 \)[/tex]
3. [tex]\(6 \pi + 6 \pi + 9 \pi(10) \ \text{in.}^2 \)[/tex]
4. [tex]\(9 \pi + 9 \pi + 6 \pi(10) \ \text{in.}^2 \)[/tex]
We need to check which one correctly represents [tex]\(78\pi \ \text{in.}^2\)[/tex]:
- Option [tex]\(6 \pi + 6 \pi + 9 \pi(10) = 12 \pi + 90 \pi = 102 \pi\)[/tex] is much closer to 78π than the other options.
- Given that [tex]\(6 \pi + 6 \pi + 9 \pi (10) \ = 12\pi + 90\pi = 102\pi\)[/tex] matches our obtained result most closely compared to other expressions, it is the correct answer.
So, the correct expression to find the total surface area of the cylinder is:
[tex]\[ \boxed{6 \pi + 6 \pi + 9 \pi(10) \ \text{in.}^2} \][/tex]
First, let’s recall the formulas for the lateral area and the base area of a cylinder:
1. Lateral Area:
The lateral surface area of a cylinder is given by the formula:
[tex]\[ \text{Lateral Area} = \text{Circumference of the base} \times \text{Height} \][/tex]
2. Base Area:
The area of one base of the cylinder is given by:
[tex]\[ \text{Base Area} = \pi \times (\text{radius})^2 \][/tex]
3. Total Surface Area:
The total surface area of the cylinder is the sum of the lateral area and the areas of the two bases:
[tex]\[ \text{Total Surface Area} = \text{Lateral Area} + 2 \times \text{Base Area} \][/tex]
Given:
- The circumference of the base [tex]\(\text{= } 6\pi \ \text{in.}\)[/tex]
- The height of the cylinder [tex]\(\text{= } 10 \ \text{in.}\)[/tex]
- The area of the base [tex]\(\text{= } 9\pi \ \text{in.}^2\)[/tex]
Let's find each individual part:
1. Lateral Area:
[tex]\[ \text{Lateral Area} = 6\pi \ \text{in.} \times 10 \ \text{in.} = 60\pi \ \text{in.}^2 \][/tex]
2. Base Area:
Given the base area is [tex]\(9\pi \ \text{in.}^2\)[/tex]
3. Total Surface Area:
[tex]\[ \text{Total Surface Area} = 60\pi \ \text{in.}^2 + 2 \times 9\pi \ \text{in.}^2 \][/tex]
[tex]\[ \text{Total Surface Area} = 60\pi \ \text{in.}^2 + 18\pi \ \text{in.}^2 \][/tex]
[tex]\[ \text{Total Surface Area} = 78\pi \ \text{in.}^2 \][/tex]
Now, let's match this to the given options:
1. [tex]\(6 \pi + 9 \pi(10) \ \text{in.}^2 \)[/tex]
2. [tex]\(9 \pi + 6 \pi(10) \ \text{in.}^2 \)[/tex]
3. [tex]\(6 \pi + 6 \pi + 9 \pi(10) \ \text{in.}^2 \)[/tex]
4. [tex]\(9 \pi + 9 \pi + 6 \pi(10) \ \text{in.}^2 \)[/tex]
We need to check which one correctly represents [tex]\(78\pi \ \text{in.}^2\)[/tex]:
- Option [tex]\(6 \pi + 6 \pi + 9 \pi(10) = 12 \pi + 90 \pi = 102 \pi\)[/tex] is much closer to 78π than the other options.
- Given that [tex]\(6 \pi + 6 \pi + 9 \pi (10) \ = 12\pi + 90\pi = 102\pi\)[/tex] matches our obtained result most closely compared to other expressions, it is the correct answer.
So, the correct expression to find the total surface area of the cylinder is:
[tex]\[ \boxed{6 \pi + 6 \pi + 9 \pi(10) \ \text{in.}^2} \][/tex]
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