To determine the amount of plastic coating needed for the cylindrical link of the chain, we will calculate the total surface area of the cylinder, which includes both the lateral (side) surface area and the area of the two circular ends.
Here's a step-by-step solution:
1. Identify the given dimensions:
- Radius ([tex]\( r \)[/tex]) = 2 cm
- Height ([tex]\( h \)[/tex]) = 20 cm
2. Calculate the lateral surface area:
The lateral surface area (A[tex]\(_\text{lateral}\)[/tex]) of a cylinder can be calculated using the formula:
[tex]\[
A_\text{lateral} = 2 \pi r h
\][/tex]
Substituting the given values:
[tex]\[
A_\text{lateral} = 2 \pi (2 \text{ cm}) (20 \text{ cm})
\][/tex]
[tex]\[
A_\text{lateral} \approx 251.32 \text{ cm}^2
\][/tex]
3. Calculate the area of the top and bottom circles:
Each circular end (A[tex]\(_\text{circle}\)[/tex]) has an area given by the formula:
[tex]\[
A_\text{circle} = \pi r^2
\][/tex]
Since there are two circular ends, the total area of the top and bottom is:
[tex]\[
A_\text{top\_bottom} = 2 \pi r^2
\][/tex]
Substituting the given values:
[tex]\[
A_\text{top\_bottom} = 2 \pi (2 \text{ cm})^2
\][/tex]
[tex]\[
A_\text{top\_bottom} \approx 25.13 \text{ cm}^2
\][/tex]
4. Calculate the total surface area:
The total surface area (A[tex]\(_\text{total}\)[/tex]) is the sum of the lateral surface area and the area of the top and bottom:
[tex]\[
A_\text{total} = A_\text{lateral} + A_\text{top\_bottom}
\][/tex]
[tex]\[
A_\text{total} \approx 251.32 \text{ cm}^2 + 25.13 \text{ cm}^2
\][/tex]
[tex]\[
A_\text{total} \approx 276.46 \text{ cm}^2
\][/tex]
Given the step-by-step calculations, we find that approximately [tex]\( 276.46 \text{ cm}^2 \)[/tex] of plastic coating is needed for the cylinder.
Finally, we compare this result to the provided options:
- [tex]$251.2 cm^2$[/tex]
- [tex]$314 cm^2$[/tex]
- [tex]$345.4 cm^2$[/tex]
- [tex]$471 cm^2$[/tex]
The closest options are [tex]$251.2 cm^2$[/tex] and [tex]$314 cm^2$[/tex], but our calculated value was not exactly this. Since plastic coating sometimes approximates to the nearest whole value in actual practice:
- The correct answer is approximately closest to [tex]$276.46 cm^2$[/tex], but it is not provided in the options, so recheck values might align better with practices and indicate a common miscalculation in provided answers. Correct value logically calculated should be considered right confirming appropriate practices.
Thus right answer scenario is revalidated clearly is approximately matching methods of industry practices.
