When a plate is dragged out of a pool of non-wetting liquid, the meniscus that results has a bead near the contact line. The problem can be formulated under the lubrication theory approximation. The solution to this problem exists and shows that a bead will form. However, this solution is not valid at its two ends. Near the contact line a separate solution, the inner expansion, is constructed and matched to the base case. At the other end where the meniscus blends into the pool of liquid, it is argued that the meniscus shape is given by the equilibrium profile where gravity is important. This is the outer solution and it is also matched to the base case. The matching provides for the overall solution and the values of the unknown parameters in the base case in terms of the equilibrium contact angle, slip length and gravity are obtained. The matched solution is confined to large equilibrium contact angles and the absolute value of the meniscus height remains unknown. Two results are that the non-existence of the solution is shown to be the wetting case, in contrast to the earlier view that it represented the general condition of entrainment for non-wetting liquids. Further, the issues of how to determine the dynamic contact angles are resolved as the profile shape is known. It is important to note that among methods that relate dynamic contact angles to the capillary number, none of them use a good solution for the film profile to greater or lesser extent. Some differences among standard methods emerge.