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An axisymmetric system of mould, glass piece, plunger and plunger cavity is considered. The state problem is given as a stationary head conduction process. The system has given heat source representing glass piece and is cooled by flowing water inside the plunger cavity and outside by environment of the mould. The design variable is taken to be the shape of inner surface of the plunger cavity. The cost functional is the second power of the norm in the weighted space $ L^2_r $ of difference of trace of temperature from given constant, which is evaluated on the outward boundary of the plunger. Existence and uniqueness of the state problem solution and existence of a solution of the optimization problem are proved.