1. Seat allocation:
a) Problem description: A total of 200 students from three departments (100 for Department A, 60 for Department B, and 40 for Department C). There are a total of 20 representative meetings, which are distributed proportionally, with 10, 6, and 4 seats in the three departments.
b) The problem exists: Now that the students change departments, the numbers of the three departments have become 103, 63, 34. How to allocate the 20 seats? In order to make it as "fair" as possible.
c) Solution: Put forward different hypotheses, discuss different methods, compare and analyze different methods (what fairness conditions are met), and draw conclusions.
How to measure fairness?
Proportional method
Relative unfairness
Q value method
Least square method to solve equation coefficients
2. The effect of double-glazed windows:
a) Problem description: How much heat loss does a double-glazed window reduce compared with a single-glazed window of the same material?
b) Problem assumption: heat transmission only has leaflets and no convection; T1 and T2 remain unchanged, and the heat conduction process is in a steady state; the material is uniform and the heat conduction process is in a steady state.
c) Modeling: heat conduction law model. (With formula)
d) Analysis: It belongs to the "test analysis" type, modeling analysis, calculation results, and conclusions.
e) Extension: Considering the actual situation, it will be better to draw a conclusion through further analysis.
3. Results of rowing competition:
a) Problem description: Comparing the results of four international championships of four types of boats (single, double, four, and eight), it is found that there is a certain relationship with the number of oarsmen. Try to establish a mathematical model to reveal this relationship.
b) It belongs to statistics and mathematical model fitting type;
c) Problem analysis: The relationship between rowing speed and the number of oars: forward resistance and forward power, etc.
d) Problem modeling: make assumptions and use appropriate physical laws to build models;
e) Model checking: the most important part! That is, through actual data, the least square method is used for model checking!
4. The purpose of the video recorder counter: mechanism analysis
a) Problem description: After testing, a video tape went from beginning to end, it took 183 minutes and 30 seconds, and the counter reading changed from 0000 to 6152.
Asked the video tape has been rotated more than half in one use, and the counter reads 4580. Can the remaining section be recorded for the next hour?
b) Requirements: Not only answer the questions, but also model the relationship between the counter reading and the transit time of the video tape;
c) Thinking: Does the counter reading increase evenly?
d) Problem analysis: Through the actual observation of the working principle of the video recorder counter, the essence of the problem is found; then the model assumption is made, and the existing physical knowledge is modeled, the parameters are determined, and then the actual model is tested!
5. Physical exchange:
a) Problem description: A has item X and B has item Y. In order to meet higher needs, the two parties agree to exchange part of each other. Research the physical exchange program.
b) Establish a two-dimensional model according to the actual situation: (x, y) means;
c) Perform different modeling processes according to different assumptions.
6. Conveyor belt efficiency: physical model
a) Problem description: The worker hangs the produced product on the empty hook above him and transports it away. If the number of workbenches is fixed, the more the number of hooks, the more products will be transported by the conveyor belt. After the product enters a steady state, it will be given To measure the efficiency of conveyor belts, and to study ways to improve the efficiency of conveyor belts.
b) Problem analysis: Make a hypothesis, and then because the measurement indicators need to be selected by themselves, it is necessary to make a decision based on the actual situation and determine the indicators.
c) Model establishment, and propose ways to improve efficiency.
7. Set sail:
a) Problem description: Sailing sailing on the sea by wind to determine the best sailing direction and sail orientation.
b) Simplify the problem: the east wind is blowing strongly on the sea, suppose the sailing boat is going from point A to the B electricity in the east, and determine the course of the sail and the direction of the sail.
c) Problem analysis: Carry out acceptance analysis, model establishment, and solution completely based on actual conditions.
to sum up:
1. Seat analysis: make different assumptions, draw different models, and conduct comparative analysis;
2. The effect of double-glazed windows: reasonable assumptions, model establishment based on existing physical knowledge, analysis of different situations, and conclusions;
3. The results of the rowing competition: reasonable assumptions, model assumptions based on existing physical knowledge, and actual data can be measured! Perform a model check! Finally draw a conclusion;
4. The purpose of the video recorder counter: "mechanism analysis" the real physical condition of the object! After that, assumptions are made, the model is established, the parameters are estimated (often using least square estimation), and finally the model is tested, and then conclusions are drawn.
5. Physical exchange: two-dimensional modeling, different assumptions, different principles, and different processing methods.
6. The efficiency of the conveyor belt: According to the actual situation and existing knowledge, make reasonable assumptions, modeling and analysis, this topic focuses on the way to "improve efficiency"!
7. Set sail for a long voyage: According to the actual situation, carry out acceptance analysis, and build and solve the model under the premise of certain reasonable assumptions.
The above lists some basic model building methods. Of course, the questions in the actual competition will not be so simple, but through them, you can still experience the actual meaning and connotation of modeling.