Let's break down the problem step by step.
### 1. Finding the Lateral Area:
The lateral (curved surface) area of a right cylinder can be calculated using the formula:
[tex]\[
\text{Lateral Area} = 2 \pi r h
\][/tex]
Given in the problem, the height [tex]\( h \)[/tex] of the cylinder is [tex]\( 2r \)[/tex]. Plugging this value into the formula for the lateral area, we get:
[tex]\[
\text{Lateral Area} = 2 \pi r (2r) = 2 \pi r \cdot 2r = 4 \pi r^2
\][/tex]
So, the lateral area of the cylinder is:
[tex]\[
4 r^2 \pi \quad \text{square inches}
\][/tex]
### 2. Finding the Area of the Two Bases:
Each base of the cylinder is a circle, and the area of a circle is given by:
[tex]\[
\text{Area of one base} = \pi r^2
\][/tex]
Since there are two bases, the total area of the two bases together is:
[tex]\[
\text{Total area of the two bases} = 2 (\pi r^2) = 2 \pi r^2
\][/tex]
So, the area of the two bases together is:
[tex]\[
2 r^2 \pi \quad \text{square inches}
\][/tex]
### 3. Finding the Total Surface Area:
The total surface area of the cylinder is the sum of the lateral area and the area of the two bases. So, we add the lateral area and the area of the two bases:
[tex]\[
\text{Total Surface Area} = \text{Lateral Area} + \text{Area of the two bases}
\][/tex]
Substitute the values we found:
[tex]\[
\text{Total Surface Area} = 4 \pi r^2 + 2 \pi r^2 = (4 \pi r^2 + 2 \pi r^2) = 6 \pi r^2
\][/tex]
So, the total surface area of the cylinder is:
[tex]\[
6 r^2 \pi \quad \text{square inches}
\][/tex]
To summarize:
1. The lateral area of the cylinder is [tex]\(4 r^2 \pi\)[/tex] square inches.
2. The area of the two bases together is [tex]\(2 r^2 \pi\)[/tex] square inches.
3. The total surface area of the cylinder is [tex]\(6 r^2 \pi\)[/tex] square inches.
