**QUESTION:-**

If the total surface area of a closed cylindrical tube is numerically twice its volume, and the base radius and height of the tube are integers, then how many possible values exist for the volume of the tube?

OPTIONS

1) 0

2) 1

3) 2

4) 3

5) None of these

**Solution**Let the base radius and height of the tube be r and h respectively.

Total surface area of the tube = curved surface area + area of the base + area of the top = 2πrh + 2πr2

Volume of the tube = πr2h

Now, total surface area of the tube is twice its volume.

∴ 2πrh + 2πr^2 = 2πr^2h

∴ r + h = rh

∴ r = h/(h – 1)

The only value of h for which the RHS is an integer is h = 2

For all other values of h, the numerator and denominator of the RHS are consecutive numbers and hence, relatively prime.

For h = 2, we get r = 2

Thus, there is only one possible value of the volume of the tube.

Hence, option 2.

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