The surface of a cylinder has three pieces: two end-caps with area (pi)r^2 and one rectangle that is wrapped around the cylinder which has area C*h where C is the circumference of the end-cap and h is the height of the cylinder. So the SA of the cylinder is

SA = 2(pi)r^2 + 2(pi)rh

The volume of the cylinder is

V = (pi)r^2 * h

So. The curved surface has an area of (1/3)*(42 cm) = 2(pi)rh.

We need r and h and we only have one equation. We need another.

Well, SA = 42 cm = 2(pi)r^2 + 2(pi)rh.

The first equation gives us a value of rh:

rh = (1/3)*(42)/[2(pi)] = 2.22817 cm^2 or so.

We can insert this value into the second equation:

42 = 2(pi)r^2 + 2(pi)(2.22817)

We can solve this for r. I get r = 2.11100 cm or so.

Now we can find h, since we know the value of rh and r:

h = (rh)/r = 1.05550 cm or so.

Now:

V = (pi)r^2 * h = 14.77703 cm^3 or so.

-Dan