First, we will find the volume of water that is in the conical cup. To do this, we will use the equation for a volume of a cone which is:

volume of a cone=(1/3)(pi)(r^2)(h)

since the diameter is 14cm, we know that the radius is half that, or 7cm. We will assume that the problem meant to say that the 18cm is the height of the cup. Plugging this information in, we get:

Volume of cup=(1/3)(pi)(7^2)(18)

solving, we get:

Volume of cup=294(pi) cm^3

Now we will solve for the height of the water when it is poured into a cylindrical vessel with a base of diameter 10 cm (i.e., a radius of 5 cm). To do this, we will use the general volume equation for that of a cylinder which is:

Volume of water in cylinder=(pi)(r^2)(h)

We know that the volume of water will stay the same, so we can use the 'volume of cup' for this amount. Plugging in what we know, we get:

294(pi)=(pi)(5^2)(h)

Solving for h, we get:

h=11.76 cm

This is the height of the water in the cylindrical vessel.