To determine the amount of plastic coating needed to coat the surface of a cylindrical link in a chain, we need to calculate the total surface area of the cylinder, including both the lateral surface area and the areas of the two circular ends.
The relevant values given are:
- Radius ([tex]\( r \)[/tex]) of the cylinder: [tex]\( 2.5 \)[/tex] cm
- Height ([tex]\( h \)[/tex]) of the cylinder: [tex]\( 22 \)[/tex] cm
- Value of [tex]\( \pi \)[/tex]: [tex]\( 3.14 \)[/tex]
### Step 1: Calculate the Lateral Surface Area
The lateral surface area of a cylinder, which is the area of the side surface (excluding the top and bottom), is given by the formula:
[tex]\[ \text{Lateral Surface Area} = 2 \pi r h \][/tex]
Plugging in the values:
[tex]\[ \text{Lateral Surface Area} = 2 \times 3.14 \times 2.5 \times 22 \][/tex]
### Step 2: Calculate the Surface Area of the Two Circular Ends
The surface area of one circular end of the cylinder is given by the formula:
[tex]\[ \text{Area of One End} = \pi r^2 \][/tex]
Since there are two ends, the total area of the two circular ends is:
[tex]\[ \text{Total Area of Ends} = 2 \pi r^2 \][/tex]
Plugging in the values:
[tex]\[ \text{Total Area of Ends} = 2 \times 3.14 \times (2.5)^2 \][/tex]
### Step 3: Calculate the Total Surface Area
The total surface area of the cylinder is the sum of the lateral surface area and the total area of the ends:
[tex]\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Total Area of Ends} \][/tex]
From the above steps, the values we obtain are:
[tex]\[ \text{Lateral Surface Area} = 345.4 \, \text{cm}^2 \][/tex]
[tex]\[ \text{Total Area of Ends} = 39.2 \, \text{cm}^2 \][/tex]
[tex]\[ \text{Total Surface Area} = 384.7 \, \text{cm}^2 \][/tex]
### Conclusion
Upon reviewing the options, the correct answer for the total amount of plastic coating needed to coat the surface of the chain link is:
[tex]\[ 345.4 \, \text{cm}^2 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{345.4 \, \text{cm}^2} \][/tex]
