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Sagot :
Let’s go through the problem step-by-step to determine which expression can be used to find the lateral area plus two bases of the cylinder. To do this, we need to calculate the total surface area of the cylinder which includes the area of the two circular bases and the lateral area (the area of the side).
### Step-by-Step Solution:
1. Calculate the Area of One Base:
- The area of one base of the cylinder is given as [tex]\( 9 \pi \, \text{in}^2 \)[/tex].
2. Calculate the Total Area of the Two Bases:
- Since there are two bases, the total area of the two bases is:
[tex]\[ 2 \times 9 \pi = 18 \pi \, \text{in}^2 \][/tex]
3. Calculate the Lateral Area:
- The circumference of the base is given as [tex]\( 6 \pi \, \text{in} \)[/tex].
- The height of the cylinder is given as 10 inches.
- The lateral area (the area of the side) can be calculated by multiplying the circumference of the base by the height of the cylinder:
[tex]\[ \text{Lateral Area} = \text{Circumference} \times \text{Height} = 6 \pi \times 10 = 60 \pi \, \text{in}^2 \][/tex]
4. Calculate the Total Surface Area (Lateral Area plus Two Bases):
- To find the total surface area, add the area of the two bases and the lateral area:
[tex]\[ \text{Total Surface Area} = 18 \pi + 60 \pi = 78 \pi \, \text{in}^2 \][/tex]
Now, let's review the provided expressions to see which one matches our calculations:
1. First Expression:
[tex]\[ 6 \pi + 9 \pi (10) \][/tex]
[tex]\[ \Rightarrow 6 \pi + 90 \pi = 96 \pi \, \text{in}^2 \quad \text{(not a match)} \][/tex]
2. Second Expression:
[tex]\[ 9 \pi + 6 \pi (10) \][/tex]
[tex]\[ \Rightarrow 9 \pi + 60 \pi = 69 \pi \, \text{in}^2 \quad \text{(not a match)} \][/tex]
3. Third Expression:
[tex]\[ 6 \pi + 6 \pi + 9 \pi (10) \][/tex]
[tex]\[ \Rightarrow 6 \pi + 6 \pi + 90 \pi = 102 \pi \, \text{in}^2 \quad \text{(not a match)} \][/tex]
4. Fourth Expression:
[tex]\[ 9 \pi + 9 \pi + 6 \pi (10) \][/tex]
[tex]\[ \Rightarrow 9 \pi + 9 \pi + 60 \pi = 78 \pi \, \text{in}^2 \quad \text{(match)} \][/tex]
Therefore, the correct expression to find the lateral area plus the area of the two bases of the cylinder is:
[tex]\[ 9 \pi + 9 \pi + 6 \pi (10) \, \text{in}^2 \][/tex]
### Step-by-Step Solution:
1. Calculate the Area of One Base:
- The area of one base of the cylinder is given as [tex]\( 9 \pi \, \text{in}^2 \)[/tex].
2. Calculate the Total Area of the Two Bases:
- Since there are two bases, the total area of the two bases is:
[tex]\[ 2 \times 9 \pi = 18 \pi \, \text{in}^2 \][/tex]
3. Calculate the Lateral Area:
- The circumference of the base is given as [tex]\( 6 \pi \, \text{in} \)[/tex].
- The height of the cylinder is given as 10 inches.
- The lateral area (the area of the side) can be calculated by multiplying the circumference of the base by the height of the cylinder:
[tex]\[ \text{Lateral Area} = \text{Circumference} \times \text{Height} = 6 \pi \times 10 = 60 \pi \, \text{in}^2 \][/tex]
4. Calculate the Total Surface Area (Lateral Area plus Two Bases):
- To find the total surface area, add the area of the two bases and the lateral area:
[tex]\[ \text{Total Surface Area} = 18 \pi + 60 \pi = 78 \pi \, \text{in}^2 \][/tex]
Now, let's review the provided expressions to see which one matches our calculations:
1. First Expression:
[tex]\[ 6 \pi + 9 \pi (10) \][/tex]
[tex]\[ \Rightarrow 6 \pi + 90 \pi = 96 \pi \, \text{in}^2 \quad \text{(not a match)} \][/tex]
2. Second Expression:
[tex]\[ 9 \pi + 6 \pi (10) \][/tex]
[tex]\[ \Rightarrow 9 \pi + 60 \pi = 69 \pi \, \text{in}^2 \quad \text{(not a match)} \][/tex]
3. Third Expression:
[tex]\[ 6 \pi + 6 \pi + 9 \pi (10) \][/tex]
[tex]\[ \Rightarrow 6 \pi + 6 \pi + 90 \pi = 102 \pi \, \text{in}^2 \quad \text{(not a match)} \][/tex]
4. Fourth Expression:
[tex]\[ 9 \pi + 9 \pi + 6 \pi (10) \][/tex]
[tex]\[ \Rightarrow 9 \pi + 9 \pi + 60 \pi = 78 \pi \, \text{in}^2 \quad \text{(match)} \][/tex]
Therefore, the correct expression to find the lateral area plus the area of the two bases of the cylinder is:
[tex]\[ 9 \pi + 9 \pi + 6 \pi (10) \, \text{in}^2 \][/tex]
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