Given the cross sectional area, CSA, of a cylinder to be
[tex]\text{CSA}=32\operatorname{cm}^2[/tex]
The cross section of a cylinder is a circle, thus the formula for the cross sectional area of a cylinder is
[tex]\text{CSA}=\pi r^2[/tex]
Solving to find the radius of the cylinder below
[tex]\begin{gathered} 32=\pi r^2 \\ \text{Divide both sides by }\pi \\ \frac{\pi r^2}{\pi}=\frac{32}{\pi} \\ r^2=\frac{32}{\pi} \\ \text{Square root of both sides} \\ \sqrt[]{r^2}=\sqrt[]{\frac{32}{\pi}} \\ r=\sqrt[]{\frac{32}{\pi}}cm \end{gathered}[/tex]
To find the total surface area, TSA, of a cylinder, the formula is
[tex]\text{TSA}=2\pi rl+2\pi r^2[/tex]
Where
[tex]\begin{gathered} l=\text{length}=25\operatorname{cm}\text{ and } \\ r=\text{radius}=\sqrt[]{\frac{32}{\pi}}cm \end{gathered}[/tex]
Subsitute the values intom the formula for the total surface area of a cylinder
[tex]\begin{gathered} \text{TSA}=2\pi rl+2\pi r^2 \\ \text{TSA}=2(\pi\times\sqrt[]{\frac{32}{\pi}}\times25)+2(\pi\times(\sqrt[]{\frac{32}{\pi}})^2) \\ \text{TSA}=501.3257+64 \\ \text{TSA}=565.3257 \\ \text{TSA}=565.3\text{cm}^2\text{ (nearest tenth)} \end{gathered}[/tex]
Hence, the total surface area of the cylinder is 565.3cm² (nearest tenth)
