To determine the amount of plastic needed to coat one link in a chain, which is in the shape of a cylinder with a radius of 3 cm and a height of 25 cm, we need to calculate the surface area of the cylinder.
The formula for the surface area of a cylinder is:
[tex]\[ \text{Surface Area} = 2\pi r h + 2\pi r^2 \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the cylinder,
- [tex]\( h \)[/tex] is the height of the cylinder,
- [tex]\( \pi \)[/tex] (pi) is approximately 3.14159.
Given the radius [tex]\( r = 3 \)[/tex] cm and the height [tex]\( h = 25 \)[/tex] cm, let's break down the calculation:
1. Calculate the lateral surface area, which is the area around the side of the cylinder.
[tex]\[ 2 \pi r h = 2 \times 3.14159 \times 3 \times 25 \][/tex]
After performing this multiplication:
[tex]\[ = 2 \times 3.14159 \times 75 \][/tex]
[tex]\[ = 471.2385 \, \text{cm}^2 \][/tex]
2. Calculate the area of the two circular bases.
[tex]\[ 2 \pi r^2 = 2 \times 3.14159 \times 3^2 \][/tex]
Simplify [tex]\( 3^2 \)[/tex]:
[tex]\[ = 9 \][/tex]
Then:
[tex]\[ = 2 \times 3.14159 \times 9 \][/tex]
[tex]\[ = 56.5487 \, \text{cm}^2 \][/tex]
3. Add the lateral surface area and the area of the two bases together to get the total surface area.
[tex]\[ \text{Surface Area} = 471.2385 \, \text{cm}^2 + 56.5487 \, \text{cm}^2 \][/tex]
[tex]\[ = 527.7872 \, \text{cm}^2 \][/tex]
Now, we compare this calculated surface area to the given choices:
- [tex]$251.2 \, \text{cm}^2$[/tex]
- [tex]$314 \, \text{cm}^2$[/tex]
- [tex]$345.4 \, \text{cm}^2$[/tex]
The closest value to our calculation of [tex]\( 527.7872 \, \text{cm}^2 \)[/tex] is:
[tex]\[ 345.4 \, \text{cm}^2 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{345.4 \, \text{cm}^2} \][/tex]
