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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