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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Prompt analysis
Surface area = 2*pi*r(r+h)

Superset
The value will be a positive real number

Translation
In order to find the answer, we need:
1# exact value of r and h.
2# Value of r^2 and rh
3# two equation 2 variables(r and h) to calculate r and h


Statement analysis
St 1: replace r or h with the term in h or r respectively and you will still be left with an expression in one variable
St 2: If you replace h with given expression, after solving, you will be left with 30pi, which is constant. ANSWER

Option B
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A right circular cylinder is a cylinder with the circular bases and with the axis joining the two centers of the bases perpendicular to the planes of the two bases.




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