Wu Zhe came back from workout and brought breakfast to the three people in the dormitory as usual.

Huang Minghai: They have all gotten up today. Yesterday several people went to bed early. Because the national prize is open today, don't think about it if you want to sleep well for the next three days. 72 hours of play is still relatively tight.

At eight o'clock in the morning, Wu Zhe's dormitory was full of four people and went out on time. All four of them also carried their own laptops.

The computer room prepared by the school for the modeling competition is not in the East Campus, but in the West Campus. So you also have to take the school bus.

The good thing is that after considering this situation, the school has set up a few temporary sleeping spots next to the computer room. You go straight in and you can fall down and sleep as long as no one is around.

Wu Zhe and the others arrived at the competition venue, it was already relatively late, and it was full of people. After all, more than 30,000 college students from 1,330 colleges and universities participated in this national award. There are more than 100 teams in the University of Science and Technology alone.

The competition is jointly organized by the Ministry of Education and the National Mathematical Society to ensure authority and fairness.

Wu Zhe, Wang Chao, Huang Minghai, and Shen Zhiwen found a place to sit down.

At nine o'clock, the bell rang.

Everyone quickly logs on to the website and downloads the questions.

The Higher Education Society Cup is composed of four questions: A, B, C, and D. The undergraduate group chooses one question from A and B to answer shortly. C and D are for the main group, which is the so-called college group.

Wu Zhe: They don't need to think about questions C and D. As long as you do either A and B, you can complete and upload the MD5 code generated by the official website software within 72 hours, and you also need to upload the corresponding pdf format paper on the end day.

Time is tight, and the topic selection will be completed within half an hour to an hour. The rest is the liver.

The question is downloaded. Wu Zhe took a look at it, and this year it was still a relatively regular topic, and there were no particularly strange topics.

Question A: Optimization of the light source of the car light line.

Question B: Mathematics in the lottery.

The first question is about the flexible use of line algebra, calculus, geometry plus an optimization problem. The second problem is statistical nonlinear optimization and evaluation. It's relatively simple.

Wu Zhe estimated that there were more people who chose the second question. The first person has a gambling nature, no matter what the lottery says, it is a gambling game. Besides, now is the time to apply what you have learned. will think about whether the winning side is big or not, and whether it is possible to win big with a small one.

Sure enough, it didn't take long to start. There were a lot more people in the room shouting for B. Wu Zhe pouted, he didn't have to think about it carefully, and the answer they came up with was that they probably won't buy lottery tickets in the future.

Wu Zhe chose question A, and he didn't bother Huang Minghai to choose the topic.

Wu Zhe now has to solve three problems: (1) under the condition of satisfying the design specification, calculate the length of the line light source to minimize the power of the line light source.

(2) For the length of the obtained light source, draw the bright area of the reflected light on the test screen in the coordinate system with a ruler.

(3) Discuss the reasonableness of the design code.

Known: its opening radius 36 mm, depth 21.6 mm. After passing through the focal point of the headlight, a uniformly distributed light source of a certain length is symmetrically placed in the horizontal direction perpendicular to the axis of symmetry.

Then take point B and point C on the same side of point A of the straight line, so that AC=2AB=2.6 meters. The light intensity at point C is required to be not less than a certain rated value (can be taken as 1 unit), and the light intensity at point B is not less than twice the rated value (only one reflection needs to be considered).

The coordinate system is established as follows, the length of the line light source is l, the power is W, B, and C, and the light intensity of the points is W and W respectively, and the expression of the sum is found first, and then the mathematical model of the whole problem is established.

The following are all in millimeters, and it is not difficult to find the equation for the reflective surface of the lamp from the information given, and the focal coordinates are (0,0,15). The energy reflected from a point light source per unit energy at point P(0,w,15) to point C(0, 2600, 25015).

Let the coordinates of the reflection point be Q, and the incident vector is recorded as the direction of the normal outside the reflection surface of the point, and it is not difficult to obtain the reflection vector, and K is the constant.

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where e is the brightness value, l is the maximum point, and is the maximum range of investigation, for example, 20mm. It also has similar properties, and the brightness and maximum points are shifted to the right accordingly. Numerical solving

Remembering its explanation as, and then finding it, it is not difficult to see that E falls in L.

This proves that e is indeed the minimum. Finally, it is calculated that e=3.294

It's a matter of finding the radiant area. The line light source and the reflective surface of the lamp are discretized into a set of point light sources and facets, and the reflected rays of each point light source about each reflective surface element of the lamp are calculated, and whether they intersect with the reflective surface of the lamp.

The collection of all these highlights is known as the Reflection Bright Zone. The upper part of the bright area is shown in the figure below (the abscissa is the x-axis, the ordinate is the y-axis, and the unit is mm), and the lower half is symmetrical with respect to the x-axis.

Another method of calculation is to establish a numerical model of the problem and solve it by numerical simulation. The specific method is as follows: after obtaining the mathematical model of the reflected light and the energy reflected on the test screen, the line light source and the reflective surface of the car lamp are discretized into a set of point light sources and face elements, and a small face element is taken near the B (or C) point of the test screen. Calculate the reflected rays of each light source for each reflective face, and superimpose the reflected energy of all the reflected rays that can reach the face, and divide by the area of the facet to be the reflected energy density of point B (or C).

Suppose the line light source is transparent. If you assume that the line light source is opaque, it seems more realistic. In this case, it is necessary to consider the occlusion of the reflected light by the line light source itself, which will be more complicated and the calculation results will be different.

At this point, the problem has actually been solved by Wu Zhe.

Next, Wu Zhe only needs to program and convert the mathematical thinking he just had just into a computer language.

Looking up at the next day, it was already meal time. Many people are just getting their hands on the idea.

Wu Zhe is not in a hurry, his progress is obviously much faster than others. glanced at Huang Minghai and them. Wang Chao was calculating there, Shen Zhiwen and Huang Minghai were watching from behind, and asked a few questions from time to time. It seems that they have also divided the work, Wang Chao on the first day, Shen Zhiwen on the second day, and Huang Minghai on the last paper.

They don't want to go to dinner anymore. Wu Zhe was about to go out to eat, so he helped them bring it over by the way.

gestured with the boss Huang Minghai, and Wu Zhe also stepped out of the computer room.

As soon as I went out, I heard someone shouting to me.

"Wu Zhe, here." Qi Ming, the secretary of the Youth League, carried the box and said.

"You're coming to the West Campus? Didn't you participate in the national awards? Wu Zhe was a little surprised.

"Well, I'm bringing you lunch, this is our class decision, although we can't play, but we have to do a good job of logistics." Qi Ming shook off his hair and said.

"Thanks, I'm going out to eat? You're welcome! Wu Zhe took the bag from Qi Ming's hand.

"Well, the top is yours, and the bottom is Wang Chao and theirs, don't forget." Qi Ming said with a smile.

"Okay, I can't forget, you go back first!" Wu Zhe raised the bag in his hand at Qi Ming.

"Okay, then I'll go, you don't have to go out to eat at night, I'll send it to you on time for the next three days."

After saying that, Qi Ming smiled and turned to leave, now Wu Zhe and they are racing against time, if they can, don't delay.