To find the surface area of a cylinder with a base radius of 2 inches and a height of 9 inches, we need to consider both the areas of the two circular bases and the area of the rectangular side that wraps around the cylinder. Here is the step-by-step solution:
1. Calculate the area of the two circular bases:
- The formula to calculate the area of a circle is [tex]\( A = \pi r^2 \)[/tex].
- For one base, with the radius [tex]\( r = 2 \)[/tex] inches:
[tex]\[ A = \pi \times (2)^2 = 4\pi \][/tex]
- There are two bases, so the combined area of the bases is:
[tex]\[ \text{Area of the two bases} = 2 \times 4\pi = 8\pi \][/tex]
- Numerically, this gives:
[tex]\[ \text{Area of the two bases} \approx 8 \times 3.14159 = 25.132741228718345 \text{ square inches} \][/tex]
2. Calculate the area of the side (rectangular part):
- The side of the cylinder, when unwrapped, forms a rectangle.
- The height of this rectangle is the height of the cylinder, [tex]\( h = 9 \)[/tex] inches.
- The width of the rectangle is the circumference of the base circle.
- The formula for the circumference of a circle is [tex]\( C = 2\pi r \)[/tex].
- With the radius [tex]\( r = 2 \)[/tex] inches:
[tex]\[ C = 2\pi \times 2 = 4\pi \][/tex]
- Therefore, the area of the side is:
[tex]\[ \text{Area of the side} = 4\pi \times 9 = 36\pi \][/tex]
- Numerically, this gives:
[tex]\[ \text{Area of the side} \approx 36 \times 3.14159 = 113.09733552923255 \text{ square inches} \][/tex]
3. Calculate the total surface area:
- The total surface area of the cylinder is the sum of the areas of the two bases and the area of the side.
- Therefore:
[tex]\[ \text{Total surface area} = 25.132741228718345 + 113.09733552923255 = 138.23007675795088 \text{ square inches} \][/tex]
In conclusion, the surface area of the cylinder with a base radius of 2 inches and a height of 9 inches is approximately 138.23 square inches.
