The answer has to be B.
Total Surface Area of Cylinder = \(2{\pi}r^2+2{\pi}rh\) (We have to find the surface area of the caps of the cylinder and then the sides)
\(2{\pi}r^2\) represents the area of the two caps
\(2{\pi}rh\) represents the area of the sides of the cylinder
To answer you specific question regarding when to use which formula, the first one i.e. \(2{\pi}rh\) is used when you are not accounting for the caps of the cylinder and the 2nd one i.e. \(=2{\pi}r^2+2{\pi}rh\) or as you put it \(=2{\pi}r*({r+h})\) is used when you are taking the caps in the surface area as well.
Statement 1: \(r=2h-\frac{2}{h}\). Just for the sake of understanding the question let's not deal with this right now.
Statement 2: \(h=\frac{15}{r}-r\)
So Area of Cylinder \(=2{\pi}r^2+2{\pi}rh\)
Substitute value of h:
So Area\(=2{\pi}r^2+2{\pi}r*(\frac{15}{r}-r)\)
So Area\(=2{\pi}r^2+2{\pi}r*(\frac{{15-r^2}}{r})\)
So Area\(=2{\pi}r^2+2{\pi}(15-r^2)\)
So Area\(=2{\pi}r^2+30{\pi}-2{\pi}r^2\)
So Area\(=30{\pi}\)
Hence B is sufficient.
Now if you use the same methodology with A you cannot end up with a reduced expression that just gives you a value. Hence A is not sufficient and B is.
Hope it helps..
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