In 2006, the National Mathematical Modeling Competition for College Students, Topic C, Excellent Paper: Optimal Design of the Shape and Size of Pop-top cans Abstract: This paper mainly considers how to design the shape and size of pop-top cans to minimize the materials used under the condition of constant volume. Firstly, the cans are measured, and the mathematical models of the second, third and fourth problems are established. The design of the optimal cans model is obtained by using LINGO software combined with the measured data. In the model 1, when the volume of a right cylindrical tank is constant, the functional relationship of material volume is established with the minimum material volume as the goal. By solving the conditional extreme value of binary function, it is found that the cylinder is the most economical when its height is twice its diameter, and its volume is 360 ml, which is basically close to the measured data with a net content of 355ml in the market. In the second model, when the volume of the cans with a right frustum on the upper part and a right cylinder on the lower part is the same, considering the least material, an optimized model is established, and the volume is still 360 ml by LINGO software, and the height of the cans is calculated to be about twice the diameter. In the third model, the bottom bracket (ring) and the arch surface with certain curvature are designed according to the stress on the beam fulcrum and the principle of arch bridge design in mechanics, considering the pressure at the tank bottom. At the same time, the ratio of diameter to height is set to 0.6 18 by using the golden section, and an optimization model with the least material is established when the volume is constant. Then the relevant data are substituted into the calculation, and it is concluded that the current design of cans is the optimal design in a sense. Key words: optimization model, nonlinear programming of tank, right cylinder and right frustum.