I. INTRODUCTION In everyday life, we always buy some drinks with packaging, such as metal packaging, plastic packaging, and paper packaging. There are also various shapes in the shape of the packaging. There are cylindrical and cuboid shapes. Other shapes, beverage packaging design contains a certain mathematical principles, to consider the cost factors for the packaging design, we must also consider the protection of the ecological environment, easy to carry and other factors, therefore, manufacturers in the design of packaging must start from the above aspects . Through the research on this subject, a more reasonable design proposal for beverage packaging was obtained. At the same time, through our personal participation, we can understand more about nature, go to society, develop scientific spirit and attitude, and improve comprehensiveness. Use the learned mathematics knowledge to solve practical problems.
Second, the contents of the study (1) the size of the beverage fixed when the shape of the different packaging materials used on the impact of the number of materials and the impact of the workload of the manufacturing container;
(2) The choice of materials for beverage packaging and the development direction of packaging.
III. Collection of relevant data On Saturdays and other after school hours, we went to supermarkets and other places to measure relevant data. Table 1: Relevant data of cans (aluminum packaging)
Variety Height (cm) Bottom Radius (cm) Volume (cm3)
Shuanglu Beer 11.9 3.15 350
Coconut milk 13.2 2.65 250
Wangzai Milk 8.5 3.2 245
Lulu Drink 16.8 3.2 500
Jianlibao 10.2 3.25 330
IV. Mathematical modeling and calculation procedures for the manufacturing cost of cans Assuming that the materials used on the top and bottom sides and sides of the cans are the same, the cost of which kind of beverage is the least, and the cost is lower with the same volume. It is the most reasonable
1 Surface area of ​​cans (Shuanglu Beer) S=2π×3.15×11.9+2π×3.152≈297.8 (cm2)
The required packaging material per cubic centimeter of beer is 297.8/350 = 0.85 (cm2)
2 Surface area of ​​cans (coconut juice) S=2π×2.65×13.2+2π×2.652≈264(cm2)
The required packaging material per cubic centimeter of coconut juice is 264/250=1.06(cm2)
3 The surface area of ​​the cans (Wangzai milk) S=2π×3.2×8.5+2π×3.22≈235(cm2)
The required packaging material per cubic centimeter of Wangzai milk is 235/245=0.96 (cm2)
4 Surface area of ​​cans (Lulu Beverage) S=2π×3.2×16.8+2π×3.22≈402 (cm2)
The packaging required for each cubic centimeter of exposed beverage is 402/500 = 0.804 (cm2)
The surface area of ​​5 cans (Jianlibao) S=2π×3.25×10.2+2π×3.252=274 (cm2)
The required packaging material per cubic centimeter of Jianlibao beverage is 274/330 = 0.83 (cm2)
From the calculation results of the above five kinds of beverages, it can be seen that, in the case of the same volume, Lulu beverages use less material than other beverages, which saves resources. So, from a resource saving perspective, what kind of design is the most reasonable?
Let the height of the cans be h, the radius of the bottom circle be r, and the volume formula of the column V =
Πr2 · h, get h = V / πr2, and the surface area of ​​the cans S = 2πr2 +2πr h, so S = 2πr2 +2V/r
According to the design requirements, the volume V is a constant, the radius r is a variable, and the surface area S is a function of r. Therefore, the problem is transformed into a mathematical problem: When the value of r is taken, the function S takes a minimum value?
From S=2πr2 +2V/r = 2πr2 + V/r + V/r≥3 =3, if and only if 2πr2 = V/r, ie r =, the can has the smallest surface area 3, at which point the can is high h = 2r. That is, when the cans are designed as equilateral cylinders, the material consumed is minimal.
V. Selection and development direction of beverage packaging materials In addition to using aluminum alloy as raw materials in the packaging of beverages in the market, there are many paper packaging and plastics used for packaging. The aluminum ingots used in the production of cans have a 97% aluminum content. More than %, the other contains traces of iron, manganese, magnesium, copper, zinc, etc.. The use of paper for packaging materials is cheaper in terms of material price, and can also protect the environment, but the paper packaging is easily damaged in the beverage. The shelf life is also far less than the length of aluminum alloy packaging, plastic (polyethylene) as the packaging material is also cheaper on the price, but a clear disadvantage is the greater impact on the environment, good stability, not easy to decompose, It is easy to cause pollution.
In recent years, the packaging industry is undergoing a process of technological innovation green packaging. It refers to packaging that can be recycled and used without causing pollution to the environment. Its connotation is the recycling of resources and environmental protection. Green packaging originated in 1987. The United Nations Environment and Development Commission issued "Our Common Future". By June 1992, the United Nations and the Environment and Development Congress adopted the Rio Declaration on Environment and Development and Agenda 21, which has set off a worldwide The ecological environment is the core of the green wave. Someone called "Green package": "Enviromental Packaging"
Fiendly package) or Ecological package. Green packaging should be: no harm to the ecological environment and human health, can be recycled and reused, can promote sustainable development of the packaging. In other words, the entire process of packaging products from raw material selection, product manufacturing, use, recycling and waste should meet the requirements of ecological and environmental protection. It includes the contents of eco-environmental protection requirements such as saving resources, energy, reduction, avoiding the generation of waste, easy recycling, recycling, incineration or degradation. This is the principle that "3R" and "1D" (Reduce reduction, Reuse reuse, Recycle recycling, and Degradable degradable) are required by packaging developed in industrial countries in the world.
6. Summary In view of the above material analysis, we believe that if plastic (polyethylene) is selected as the raw material for the packaging design material of beverages, although the cost is low, due to the research on its degradability, it is still not enough. The impact is great, so it should not be used. Instead, it should choose recyclable metals or papers that are easier to decompose. It is more appropriate to use packaging materials, and it is more appropriate to choose the equilateral cylindrical shape in appearance.
Instructor: Liu Weijie Member of Project Research Group: Zhu Haixia Jiang Qingqing Wu Shufang Zheng Jingyu Fu Jiyong Lin Jiannan Zheng Xingxing Zhang Zhen
Second, the contents of the study (1) the size of the beverage fixed when the shape of the different packaging materials used on the impact of the number of materials and the impact of the workload of the manufacturing container;
(2) The choice of materials for beverage packaging and the development direction of packaging.
III. Collection of relevant data On Saturdays and other after school hours, we went to supermarkets and other places to measure relevant data. Table 1: Relevant data of cans (aluminum packaging)
Variety Height (cm) Bottom Radius (cm) Volume (cm3)
Shuanglu Beer 11.9 3.15 350
Coconut milk 13.2 2.65 250
Wangzai Milk 8.5 3.2 245
Lulu Drink 16.8 3.2 500
Jianlibao 10.2 3.25 330
IV. Mathematical modeling and calculation procedures for the manufacturing cost of cans Assuming that the materials used on the top and bottom sides and sides of the cans are the same, the cost of which kind of beverage is the least, and the cost is lower with the same volume. It is the most reasonable
1 Surface area of ​​cans (Shuanglu Beer) S=2π×3.15×11.9+2π×3.152≈297.8 (cm2)
The required packaging material per cubic centimeter of beer is 297.8/350 = 0.85 (cm2)
2 Surface area of ​​cans (coconut juice) S=2π×2.65×13.2+2π×2.652≈264(cm2)
The required packaging material per cubic centimeter of coconut juice is 264/250=1.06(cm2)
3 The surface area of ​​the cans (Wangzai milk) S=2π×3.2×8.5+2π×3.22≈235(cm2)
The required packaging material per cubic centimeter of Wangzai milk is 235/245=0.96 (cm2)
4 Surface area of ​​cans (Lulu Beverage) S=2π×3.2×16.8+2π×3.22≈402 (cm2)
The packaging required for each cubic centimeter of exposed beverage is 402/500 = 0.804 (cm2)
The surface area of ​​5 cans (Jianlibao) S=2π×3.25×10.2+2π×3.252=274 (cm2)
The required packaging material per cubic centimeter of Jianlibao beverage is 274/330 = 0.83 (cm2)
From the calculation results of the above five kinds of beverages, it can be seen that, in the case of the same volume, Lulu beverages use less material than other beverages, which saves resources. So, from a resource saving perspective, what kind of design is the most reasonable?
Let the height of the cans be h, the radius of the bottom circle be r, and the volume formula of the column V =
Πr2 · h, get h = V / πr2, and the surface area of ​​the cans S = 2πr2 +2πr h, so S = 2πr2 +2V/r
According to the design requirements, the volume V is a constant, the radius r is a variable, and the surface area S is a function of r. Therefore, the problem is transformed into a mathematical problem: When the value of r is taken, the function S takes a minimum value?
From S=2πr2 +2V/r = 2πr2 + V/r + V/r≥3 =3, if and only if 2πr2 = V/r, ie r =, the can has the smallest surface area 3, at which point the can is high h = 2r. That is, when the cans are designed as equilateral cylinders, the material consumed is minimal.
V. Selection and development direction of beverage packaging materials In addition to using aluminum alloy as raw materials in the packaging of beverages in the market, there are many paper packaging and plastics used for packaging. The aluminum ingots used in the production of cans have a 97% aluminum content. More than %, the other contains traces of iron, manganese, magnesium, copper, zinc, etc.. The use of paper for packaging materials is cheaper in terms of material price, and can also protect the environment, but the paper packaging is easily damaged in the beverage. The shelf life is also far less than the length of aluminum alloy packaging, plastic (polyethylene) as the packaging material is also cheaper on the price, but a clear disadvantage is the greater impact on the environment, good stability, not easy to decompose, It is easy to cause pollution.
In recent years, the packaging industry is undergoing a process of technological innovation green packaging. It refers to packaging that can be recycled and used without causing pollution to the environment. Its connotation is the recycling of resources and environmental protection. Green packaging originated in 1987. The United Nations Environment and Development Commission issued "Our Common Future". By June 1992, the United Nations and the Environment and Development Congress adopted the Rio Declaration on Environment and Development and Agenda 21, which has set off a worldwide The ecological environment is the core of the green wave. Someone called "Green package": "Enviromental Packaging"
Fiendly package) or Ecological package. Green packaging should be: no harm to the ecological environment and human health, can be recycled and reused, can promote sustainable development of the packaging. In other words, the entire process of packaging products from raw material selection, product manufacturing, use, recycling and waste should meet the requirements of ecological and environmental protection. It includes the contents of eco-environmental protection requirements such as saving resources, energy, reduction, avoiding the generation of waste, easy recycling, recycling, incineration or degradation. This is the principle that "3R" and "1D" (Reduce reduction, Reuse reuse, Recycle recycling, and Degradable degradable) are required by packaging developed in industrial countries in the world.
6. Summary In view of the above material analysis, we believe that if plastic (polyethylene) is selected as the raw material for the packaging design material of beverages, although the cost is low, due to the research on its degradability, it is still not enough. The impact is great, so it should not be used. Instead, it should choose recyclable metals or papers that are easier to decompose. It is more appropriate to use packaging materials, and it is more appropriate to choose the equilateral cylindrical shape in appearance.
Instructor: Liu Weijie Member of Project Research Group: Zhu Haixia Jiang Qingqing Wu Shufang Zheng Jingyu Fu Jiyong Lin Jiannan Zheng Xingxing Zhang Zhen