B: thickness of tank wall;

B 1: the thickness of the top cover;

B2: thickness of the bottom cover;

R: the inner diameter of the middle cylinder of the tank;

R 1: the radius of the top cover;

R2: radius of the bottom cover;

H 1: the vertical distance from the top of tank to the bottom of frustum;

H2: the vertical distance from the tank bottom to the bottom of the cylindrical part;

H3: the arch height of the bottom cover of the can;

A: the total volume of materials used to make cans;

V: the volume of canned drinks (since the radius and height are much larger than the material thickness of cans, the volume of cans can be regarded as volume);

Secondly, the model assumes that (1) cans are undamaged Coca-Cola beverage cans with a net content of 355ml.

(2) The influence of temperature on the shape and size design of cans is not considered;

(3) The influence of gas pressure in the tank on the shape and size design of the tank is not considered;

(4) regardless of the length l of the seam edge;

(5) The length dimension is millimeters.

Third, model analysis, establishment and solution

Take an undamaged Coca-Cola beverage can with a net content of 355ml, and measure the data of each part of the can with a caliper of one thousandth. We think these data are needed to verify the model. The measured data are shown in table 1. Table 1 is as follows:

(1) Measure the data needed to verify the model.

Average value of Coca-Cola cans at the inspection site (unit: mm)

Total height of tank (h? ) 122.90 top cover thickness of cans (? b 1)？ 0.3 1? Thickness of tank bottom cover (? b2)0.30

Thickness of tank wall (b? )? 0. 15? The radius of the cylinder in the middle of the jar (r? )3 1.75? Radius of can top cover (r 1? )29.07

Radius of tank bottom cover (? r2)26.75？ The vertical distance from the top of the jar to the bottom of the frustum (? 13.00

The vertical distance from the bottom of the tank to the bottom of the cylindrical part (h2? ) 7.30 tank bottom arch height (h3? ) 10. 10

② Optimal model when cans are straight: Assume that beverage cans are straight, as shown in the figure.

In fact, due to the requirements of manufacturing technology, it is impossible to be a perfect cylinder in mathematics, but it is approximately reasonable to simplify the problem in this way. When the volume of a beverage can is required to be constant, the ratio of the diameter of the top cover to the height from the top cover to the bottom can minimize the materials used to manufacture the can. Under this simplification, there is obviously R? =r 1？ =r2？ , starting from the hypothesis? V=πr2h？ .