Let's solve the problem step-by-step.
### Given Information:
1. Circumference of the base of the cylinder (C) = 22 inches
2. Sum of the radius (r) and height (h) of the cylinder = 15 inches
### Step-by-Step Solution:
#### Step 1: Calculate the Radius
The formula for the circumference of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
We are given the circumference (C) as 22 inches. Solving for the radius (r), we get:
[tex]\[ 22 = 2 \pi r \][/tex]
Rearranging for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{22}{2 \pi} \][/tex]
#### Step 2: Calculate the Height
We are given that the sum of the radius and the height is 15 inches:
[tex]\[ r + h = 15 \][/tex]
Given the value of [tex]\( r \)[/tex] from Step 1:
[tex]\[ h = 15 - r \][/tex]
#### Step 3: Calculate the Curved Surface Area (CSA)
The formula for the curved surface area of a cylinder is:
[tex]\[ \text{CSA} = 2 \pi r h \][/tex]
#### Step 4: Calculate the Base Area
The area of the base of the cylinder (which is a circle) is:
[tex]\[ \text{Base Area} = \pi r^2 \][/tex]
#### Step 5: Calculate the Total Surface Area (TSA)
The total surface area of a cylinder is the sum of the curved surface area and the areas of the two bases:
[tex]\[ \text{TSA} = 2 \pi r h + 2 (\pi r^2) \][/tex]
[tex]\[ \text{TSA} = \text{CSA} + 2 (\text{Base Area}) \][/tex]
### Final Answer with Values:
- The radius (r) is approximately 3.501 inches.
- The height (h) is approximately 11.499 inches.
- The curved surface area (CSA) is approximately 252.969 square inches.
- The total surface area (TSA) is exactly 330 square inches.
Thus, the answers are:
- Curved Surface Area: Approximately 252.969 square inches
- Total Surface Area: Exactly 330 square inches
