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Answer :
Sure, I'd be happy to help you understand how you can find the curved surface area (CSA) and the total surface area (TSA) of a cylinder with the given dimensions.
1. Given Information:
- The circumference of the base of the cylinder is 176 cm.
- The height of the cylinder is 30 cm.
2. Find the Radius:
The circumference [tex]\( C \)[/tex] of the base is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base. We can rearrange this formula to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{C}{2 \pi} = \frac{176}{2 \times \pi} \approx 28.01 \text{ cm}
\][/tex]
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]
Plug in the values [tex]\( r \approx 28.01 \)[/tex] cm and [tex]\( h = 30 \)[/tex] cm:
[tex]\[
\text{CSA} \approx 2 \times \pi \times 28.01 \times 30 \approx 5280.0 \text{ cm}^2
\][/tex]
4. Calculate the Total Surface Area (TSA):
The total surface area of a cylinder includes both the curved surface area and the areas of the two circular bases. The formula for TSA is:
[tex]\[
\text{TSA} = 2 \pi r (r + h)
\][/tex]
Substitute [tex]\( r \approx 28.01 \)[/tex] cm and [tex]\( h = 30 \)[/tex] cm:
[tex]\[
\text{TSA} \approx 2 \times \pi \times 28.01 \times (28.01 + 30) \approx 10209.98 \text{ cm}^2
\][/tex]
Therefore, the curved surface area of the cylinder is approximately 5280.0 cm² and the total surface area is approximately 10209.98 cm².
1. Given Information:
- The circumference of the base of the cylinder is 176 cm.
- The height of the cylinder is 30 cm.
2. Find the Radius:
The circumference [tex]\( C \)[/tex] of the base is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base. We can rearrange this formula to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{C}{2 \pi} = \frac{176}{2 \times \pi} \approx 28.01 \text{ cm}
\][/tex]
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]
Plug in the values [tex]\( r \approx 28.01 \)[/tex] cm and [tex]\( h = 30 \)[/tex] cm:
[tex]\[
\text{CSA} \approx 2 \times \pi \times 28.01 \times 30 \approx 5280.0 \text{ cm}^2
\][/tex]
4. Calculate the Total Surface Area (TSA):
The total surface area of a cylinder includes both the curved surface area and the areas of the two circular bases. The formula for TSA is:
[tex]\[
\text{TSA} = 2 \pi r (r + h)
\][/tex]
Substitute [tex]\( r \approx 28.01 \)[/tex] cm and [tex]\( h = 30 \)[/tex] cm:
[tex]\[
\text{TSA} \approx 2 \times \pi \times 28.01 \times (28.01 + 30) \approx 10209.98 \text{ cm}^2
\][/tex]
Therefore, the curved surface area of the cylinder is approximately 5280.0 cm² and the total surface area is approximately 10209.98 cm².
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