Certainly! Let's find the surface area of a right cylinder with a radius of 3 units and a height of 12 units using the formula for the surface area of a cylinder.
The surface area \( S \) of a right cylinder can be calculated with the formula:
[tex]\[
S = 2 \pi r h + 2 \pi r^2
\][/tex]
where \( r \) is the radius and \( h \) is the height of the cylinder.
Step-by-step Solution:
1. Identify the radius and height:
- Radius, \( r = 3 \) units
- Height, \( h = 12 \) units
2. Calculate the lateral surface area:
The lateral surface area of a cylinder is given by:
[tex]\[
\text{Lateral Surface Area} = 2 \pi r h
\][/tex]
Substituting in the values:
[tex]\[
2 \pi \times 3 \times 12 = 72 \pi
\][/tex]
So, the lateral surface area is \( 72 \pi \) square units.
3. Calculate the area of the two circular bases:
Each base of the cylinder is a circle with area given by:
[tex]\[
\text{Area of one base} = \pi r^2
\][/tex]
Since there are two bases:
[tex]\[
\text{Total Base Area} = 2 \pi r^2 = 2 \pi \times 3^2 = 2 \pi \times 9 = 18 \pi
\][/tex]
4. Calculate the total surface area:
The total surface area is the sum of the lateral surface area and the area of the two bases:
[tex]\[
S = 72 \pi + 18 \pi = 90 \pi
\][/tex]
Therefore, the surface area of the cylinder is \( 90 \pi \) square units.
So, the correct answer is:
D. [tex]\(90 \pi\)[/tex] units[tex]\(^2\)[/tex]
