To find the surface area of a cylindrical solid with a given radius and height, we need to utilize the formula for the surface area of a cylinder. The formula for the surface area [tex]\(A\)[/tex] of a cylinder is:
[tex]\[ A = 2\pi rh + 2\pi r^2 \][/tex]
where:
- [tex]\(r\)[/tex] is the radius of the base of the cylinder.
- [tex]\(h\)[/tex] is the height of the cylinder.
- [tex]\(\pi\)[/tex] is a constant approximately equal to 3.14159.
Given:
- Radius [tex]\(r = 7.7 \text{ cm} \)[/tex]
- Height [tex]\(h = 12 \text{ cm} \)[/tex]
First, we need to convert the radius and height from centimeters to meters since the final answer needs to be in square meters.
1. Convert the radius from centimeters to meters:
[tex]\[ r = 7.7 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = 0.077 \text{ m} \][/tex]
2. Convert the height from centimeters to meters:
[tex]\[ h = 12 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = 0.12 \text{ m} \][/tex]
Now substitute these values into the surface area formula:
[tex]\[ A = 2\pi rh + 2\pi r^2 \][/tex]
[tex]\[ A = 2\pi (0.077 \text{ m}) (0.12 \text{ m}) + 2\pi (0.077 \text{ m})^2 \][/tex]
Calculate each part separately:
a. The lateral surface area [tex]\(2\pi rh\)[/tex]:
[tex]\[ 2\pi (0.077 \text{ m}) (0.12 \text{ m}) = 0.058 \pi \text{ m}^2 \][/tex]
b. The area of the two bases [tex]\(2\pi r^2\)[/tex]:
[tex]\[ 2\pi (0.077 \text{ m})^2 = 0.0118 \pi \text{ m}^2 \][/tex]
Add these two areas together:
[tex]\[ A = 0.058 \pi \text{ m}^2 + 0.0118 \pi \text{ m}^2 = 0.0698 \pi \text{ m}^2 \][/tex]
Finally, simplify the total surface area:
[tex]\[ A \approx 0.0698 \times 3.14159 \text{ m}^2 \approx 0.219 \text{ m}^2 \][/tex]
Therefore, the surface area of the cylindrical solid is approximately [tex]\(0.0953 \text{ m}^2\)[/tex].
