Classmate Yan Tengfei just asked Li Xuan a math question that was more of a brain teaser, but he didn't expect Li Xuan to connect a small problem to a big one.

The sphere Li Xuan drew on the draft was marked with the sun's rays, equator, earth's axis, morning and evening lines, and the letters nsmbo. Yan Tengfei looked at it curiously and couldn't help but ask: "Master Li Xuan, what are you doing?"

Li Xuan explained: "I was thinking about how many days it would take for the snail to crawl out when the crawling speed of the snail is the same at night and during the day. Because I thought that the time of night and day is different around the world, I wanted to calculate the time of night and day."

"Are there any results?" Yan Tengfei could only admire him at the moment. He hadn't learned the twilight line yet in high school geography, so he felt profound when looking at Li Xuan's calculation process.

"There are results, we've calculated them." Li Xuan nodded.

Using the properties of the sphere, the sun's rays passing through the direct point must pass through the center of the earth and be perpendicular to the dusk line. The triangular relationship for calculating day length can be obtained. The main relationship is obtained. The other work is to simplify the calculation.

At this time, Liang Zhihui stood next to Li Xuan, saw clearly the whole process of Li Xuan's calculation, and was also thinking about it. Finally, he said: "Li Xuan, your formula is a bit complicated and should be simplified."

Li Xuan is also considering simplification.

This formula does not conform to the simple beauty of mathematics describing the laws of nature. The mathematical formulas in physics textbooks are all very simple. Li Xuan considered combining the two unknown quantities into one unknown quantity and returning to the reference earth sphere in the painting.

Finally he found that it seemed like it could really be simplified.

First introduce the local noon sun height h.

According to geometric relations, the local noon sun height h is equal to the local latitude minus the latitude of the direct sun point plus 90°.

After substitution and simplification, the expression form of the day length formula can be changed to f=2/15°os(tanφcot(φ+h)).

This is very concise.

The day length problem was solved and became a quantitative analysis. The laws of nature always appear to be very simple in mathematics. Li Xuan felt amazing. Various equations in physics are also very simple. Is it a coincidence?

He can't understand it, but it doesn't prevent Li Xuan from experiencing the beauty of science.

The only question now is whether the formula is correct?

Is it right?

Li Xuan didn't know, so he decided to verify it.

In a special case, the latitude on the equator is 0°. The unknown quantity is calculated based on the geometric relationship, and the specific value is then entered into the formula. It can be calculated that the day length is 12 hours.

In another special case, the latitude of Yangcheng is 22°. When calculating the summer solstice and winter solstice, the formula can calculate that the length of day and night is exactly the opposite.

Finally, in special circumstances, the day length in the Arctic and Antarctic circles is on the summer solstice, and the calculation result is also 0 hours.

Li Xuan verified the three special situations, which were consistent with the actual situation, and found no problems with the formula. He then looked at the process of establishing the formula and stipulated the value range of the unknown quantity.

All problems are solved.

Liang Zhihui looked at Li Xuan's computing project several times and found no faults. His heart was filled with admiration and unspeakable sadness. The longer he spent time with Li Xuan, the more he felt inferior to Li Xuan. This feeling was not pleasant.

Yan Tengfei was really convinced by Li Xuan: "Master, you are so awesome. If your formula is correct and the most important relationship is found, the problem of snail crawling will be simple. But if the snail crawls up during the day

It makes sense that the speed is different from climbing down at night."

Li Xuan smiled: "Yeah, but the difference in speed is not the key. As long as the time between day and night is calculated, the remaining problem is just a matter of speed multiplied by time. The difficult part is the time between night and day.

"

"Too."

Yan Tengfei didn't know what to say. Compared with Li Xuan, he felt like a child still playing in the mud in mathematics. Li Xuan was already at the stage of finding problems to improve himself.

…

Not long after the evening self-study started, Zhao Dongming walked into the self-study room of the mathematics competition group. He was actually passing by to see if the students had any questions. When he came in, he saw Yan Tengfei and Liang Zhihui surrounding Li Xuan.

The students walked around freely during self-study, but Zhao Dongming didn't care. It was a good phenomenon to discuss problems in the competition class. The students knew that they asked Li Xuan, the most powerful person in mathematics, for help, and he was too late to be happy.

Zhao Dongming walked over, a little strange: "What's wrong?"

"Teacher, Li Xuan is so awesome." Yan Tengfei was a little excited, "He actually calculated the formula for calculating the length of day and night based on a mathematical question I asked."

Zhao Dongming was slightly startled. Hearing what Yan Tengfei said, he understood what happened. Finally, he picked up Li Xuan's draft paper and was dumbfounded.

Li Xuan's ability to think divergently and build mathematical models shocked him. He had studied mathematics for many years, but he had never used mathematics as a tool to build mathematical models to solve such problems like Li Xuan.

Zhao Dongming sighed with emotion: "That's a good calculation. This is how you should learn mathematics, and solve problems in life."

What is the use of this formula?

With this formula, write a small program. As long as you can input the latitude of your location and query the latitude of the direct sun point through the date, the day and night time can be directly output by the program.

Or it could be more high-tech and provide a positioning system to directly determine the local latitude. Then you can know when it will be bright and dark locally. If you want to know when to watch the sunrise, you don’t have to wait in advance, just seize the time.

Just fine.

The more Zhao Dongming thought about it, the more wonderful the formula became. After a while, he suddenly breathed a sigh of relief and said, "Li Xuan, please write a small paper and submit it to a mathematics newspaper."

"Ah?" Li Xuan was surprised, "Teacher, this kind of problem can be solved by high school students, but the magazine will accept it?"

Yan Pengfei, who is a high school student, listened: "..."

Zhao Dongming smiled and said: "Of course I will accept it. Your formula is very interesting. The paper does not have to be high-level and difficult for ordinary people to understand. Some mathematics magazines like to accept these papers that are closely related to real life.

Mathematics paper.”

Li Xuan: "Then how should I write it?"

Zhao Dongming: "Just write about the derivation of the formula, the meaning of the formula, and the application of the formula. Let's just talk nonsense and make up the word count to 3,000 words. Of course, before that, think of a title first."

Li Xuan thought for a moment and said straightforwardly: "Teacher, the topic is: On whether a snail can climb out of the well when it climbs up during the day and down at night?"

Liang Zhihui: "..."

Yan Tengfei shook his head: "No, it would be better to have a shorter title, for example, can a snail climb out of a well?"

Zhao Dongming choked and said angrily: "Remove the snail from this paper! Who cares if the snail climbs up and down, whether it is the loss of the snail's morality or the distortion of the snail's nature? It doesn't matter. This paper of yours

, you are not allowed to write a single word about snail, and the title is called "Formula for Calculating Day and Night Time Around the World."

"Teacher, I feel like writing a paper is very troublesome, why don't we just forget about it?" Li Xuan didn't want to start writing. If he had the time, he might as well do one more math problem.

Zhao Dongming was silent for a moment and said: "There is a royalties."

"Okay."

Li Xuan picked up the pen and started writing in an instant, "Teacher, wait a moment, I will finish it immediately in the evening."
Chapter completed!