(This article won the first prize of the first "Cradle Cup"-"Mathematics in Life" composition competition for junior middle school students in Wenzhou)
On 1 night during the Eleventh Period, on my way back to Yongqiang from Wenzhou, I passed a tunnel (Zhu Mao Tunnel under Bailou Building). When the car sped through the tunnel, I found that the cross section of the highway tunnel was a rectangle under the arch ring, and almost all the cross sections I saw were of this shape. Why is the section shape of highway tunnel different from other shapes? So I decided to use mathematical knowledge to calculate and study the relationship between the cross-section shape of highway tunnel and the effective traffic area and cross-section perimeter (directly related to the cost of manufacturing materials), trying to find a more reasonable and economical cross-section shape of tunnel.
First, design highway tunnels with different cross-sectional shapes.
For the convenience of calculation, I set the effective traffic area to 4m× 4m, and the highest section of the tunnel is 6m.
Figure ① Semicircle plus Square Figure ② Triangle plus Square Figure ③ Trapezoid plus Square
Figure ④ Square plus rectangle Figure ⑤ Square
2. Calculate the total cross-sectional area, cross-sectional perimeter and actual area ratio of tunnels with different shapes.
Actual area ratio of tunnel = effective traffic area/total area of tunnel section = 16 m2/ total area of tunnel section.
The first figure: (semicircle plus square)
Total area of tunnel section = effective traffic area+semicircle area =16m2+6.28m2 = 22.28m2.
The actual area ratio of the tunnel in this drawing is =16m2/22.28m2 ≈ 71.8%.
Circumference of tunnel section in this drawing = 3× 4m+п r =12m+6.28m =18.28m.
The second figure: (triangle plus square)
Total area of tunnel section = effective traffic area+triangular area = 16m2 +4m2 =20m2.
The actual area ratio of the tunnel in this drawing = 16m2/20m2 = 80%.
The tunnel section perimeter in this drawing is ≈ 3× 4m+2× 2.83m =12m+5.66m =17.66m.
The third figure: (trapezoid plus square)
Total area of tunnel section = effective traffic area+trapezoidal area = 16m2 +6m2=22m2.
The actual area ratio of the tunnel in this drawing =16m2/12m2 ≈ 72.7%.
The tunnel section perimeter in this drawing is ≈ 3× 4m+2× 2.24m+2m =12m+6.48m =18.48m.
The fourth figure: (square plus rectangle)
Total area of tunnel section = rectangular area 1 =4m×6m=24m2.
The actual area ratio of the tunnel in this drawing = 16m2/24m2 ≈ 66.7%.
Circumference of tunnel section in this drawing =(4m+6m)×2=20m.
The fifth picture: (square)
Total area of tunnel section = rectangle 2 area =4m×4m= 16m2.
The actual area ratio of the tunnel in this drawing =16m2/16m2 =100%.
Circumference of tunnel section in this drawing =4m×4m= 16m.
Comparison of total cross-sectional area, cross-sectional perimeter and actual area ratio of tunnels with different shapes
FigureNo. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5
The total cross-sectional area is 22.28m2 20m2 22m2 24m2 16m2.
The use area ratio is 71.8% 80% 72.7% 66.7%100%.
The section perimeter is18.28m17.66m18.48m20m16m.
Thirdly, the analysis and research of the calculation results.
From the calculation results, it is concluded that 1 and the practical area ratio of different tunnel sections have certain correlation with the section perimeter, that is, the higher the practical area ratio, the smaller the perimeter (the most material saving). 2. The tunnel with the cross-sectional shapes of Figure 5 and Figure 2 has a high practical area ratio and the least manufacturing materials.
Why don't common tunnel sections adopt the shapes shown in Figure 5 and Figure 2? Instead, adopt the tunnel section shape as shown in figure 1 So I tried to find the reason online.
Inch net/index.php? Title =% E9% 9A% A7% E9% 81%93&; Variant=zh-cn learned some knowledge about tunnel structure. Tunnel body is the main part of tunnel structure and the passage for vehicles. Lining-a permanent support for bearing stratum pressure, maintaining rock mass stability and preventing stratum deformation around the tunnel. It consists of arch ring, side wall, joist and inverted arch. The arch ring is located at the top of the tunnel and is semicircular, which is the main part to bear the formation pressure. Side walls are located on both sides of the tunnel and bear the earth pressure from the arch ring and the side of the tunnel. Sidewalls can be divided into vertical and curved shapes. The joist is located between the arch wall and the side wall, and is used to support the arch ring to prevent it from loosening and cracking when the bottom of the arch ring is hollowed out. Inverted arch is located at the bottom of the pit, and its shape is similar to that of ordinary arch ring, but its bending direction is opposite to that of arch ring, which is used to resist soil sliding and prevent the bottom soil from rising.
This data shows that the number of 1 is usually used for tunnel cross section, mainly considering the pressure of confined strata to make the tunnel structure more firm and safe.
Why not use Figure 2? I have been unable to find relevant and effective information. I think it may be related to the firmness or visual effect of the structure, and the difficulty of tunnel engineering or other reasons need further study. If there is not much difference between Figure 1 and Figure 2 in these respects, I suggest that Figure 2 be adopted, because the tunnel with this shape has a high practical area ratio and the least materials for manufacturing.
Cross-scientific trend of angle calculation in mathematics
(This article won the second prize in the second "Cradle Cup" junior high school students' mathematics thesis appraisal in Wenzhou)
There are many ways to calculate angles in mathematics. So far, we have learned the proof of congruence of triangle, equilateral triangle and isosceles triangle, and the content of parallel lines in the first chapter of the first volume of Grade 8. But I got bored when I was doing 1 1 in the first chapter!
1, original title:
In billiards, when the cue ball moves, if the cue ball P touches the point A near the table, bounces off the table and touches the point B near another table, and then bounces off, is the route BC that the cue ball P passes parallel to the PA?
As shown in figure 1, it is almost difficult to solve problems with conventional mathematical problem-solving ideas. I have pondered it for a long time and discussed it with several classmates, but there is still no good solution. Even we are wondering if this topic is wrong, so we confidently find a teacher and ask the solution to this problem. The teacher told us that the method is:
Solution: According to the principle of plane mirror reflection in physics (the reflection angle is equal to the incident angle), it is known that ∠2=∠ 1, ∠4=∠3,
∠∠2 and ∠3 are complementary ∴∠ 1+∠ 2+∠ 3+∠ 4 = 180.
∵∠ 1+∠2+∠3+∠4+∠5+∠6=360
∴∠5+∠6= 180
∴PA‖CB (the inner angles on the same side are complementary and the two straight lines are parallel)
I was shocked. It is incredible that there is interdisciplinary knowledge to solve mathematical problems according to the principle of plane mirror reflection in science. The teacher said yes, and I was puzzled.
2, the operation of interdisciplinary problems in the mathematics corner of the senior high school entrance examination:
Why does physical knowledge appear in the calculation of mathematical angle? I started to investigate and search, but I was still surprised. It turns out that the proposition of senior high school entrance examination has the trend of interdisciplinary comprehensive questions.
① (Yancheng City, Jiangsu Province, 2002) As shown in Figure 2, the light L shines on the plane mirror I, and then reflects back and forth between the plane mirror I and I and II. It is known that ∠ α = 55 and ∠ γ = 75 are ∠β.
Solution: According to the principle of plane mirror reflection in physics (the reflection angle is equal to the incident angle), we can get:
∠BAC=∠α=55，∠CBA=∠γ=75
∴∠bca= 180-∠BAC-∠CBA = 180- 130 = 50
We can get ∠ ACN = ∠ BCN = ∠ CAN = 25 from the "normal" knowledge in physics.
And ∵∠ BCN+∠ β = 90.
∴∠β=90 -∠BCN=65
② As shown in Figure 3, the angle of intersection between plane mirror α and β is θ, the incident light AO is parallel to β, and the reflected light O'b is parallel to α after two reflections. What is ∠ θ?
Solution: ∫BO '‖α
∴∠∠ 1 = ∠ 2 (two straight lines are parallel and have the same angle)
And ∠3=∠4 (two straight lines are parallel and the internal dislocation angles are equal).
∫AOβ
∴∠∠ 1 = ∠ 5 (two straight lines are parallel with the same included angle),
According to the principle of plane mirror reflection in physics (reflection angle equals incident angle):
∠2=∠3,∠5=∠6,
∴ Get: ∠ 1=∠2=∠3=∠4=∠5=∠6.
∵∠4+∠5+∠6= 180 ∴∠4=∠5=∠6=60
∴∠ 1=∠2=∠3=∠4=∠5=∠6=60
∵∠3+∠6+∠θ= 180 ∴∠θ= 180 -∠3-∠6=60
From the process of solving the above problems, we can easily find that both the angle calculation in ordinary life and the angle calculation in the senior high school entrance examination mathematics have partially infiltrated the scientific content, especially the optical knowledge, so that the problems that were difficult to solve with pure mathematical knowledge can be successfully solved with the help of science. Yes, this shows that cross-professional comprehensive topics have now become a new trend in the proposition of the senior high school entrance examination.
3. Analyze the reasons and their influence on modern students:
Why is there such a comprehensive question? It is actually very simple to think about it carefully, because solving practical problems with mathematical knowledge is the starting point of learning mathematics, and when practical problems are difficult to be solved with pure mathematics, the connectivity of disciplines naturally becomes the inevitable path to solve problems. It is not difficult to imagine how common and important it will be to solve more practical problems across disciplines in a more complex world in the future.
But for us students, this trend is undoubtedly a new great challenge, which is the continuity of the discipline and the chain of thinking. This is what modern students need more than former students. This will be a challenge, and the mindset will be an extinction, such as the above three typical examples. It will be quite difficult and time-consuming if a student only wants to solve it with pure mathematical thinking instead of thinking with more eyes.
4. Summarize and put forward my opinions and suggestions:
From the calculation of the angle in the textbook to the calculation of the angle in today's senior high school entrance examination, I was at a loss at first. After searching and analyzing, I finally understand that this is a trend of the senior high school entrance examination proposition, and it is also a new problem-solving idea and method due to the improvement of the application scope of mathematics in life.
I was surprised and happy with my discovery. Fortunately, I found such a problem. I believe I will be more careful in solving math problems in the future. What should I do if I don't misuse the knowledge of different subjects and cause unnecessary points? This is a pity, but it is indeed a realistic problem for us now, so I put forward the following suggestions and my views:
(1) The all-round development of the discipline encounters interdisciplinary comprehensive problems, and it is absolutely not allowed to be partial. Only students with comprehensive subjects will have a higher winning rate. After all, the knowledge of two or more subjects is only the score of one subject, which is regrettable because of the lack of another subject.
(2) Do more questions, accumulate experience and do more questions. You will become sensitive to these types of questions and your thinking will be unimpeded, so experience is very important. If you do too much, you will naturally think of which subjects to use when you see the comprehensive questions.
(3) Although we should pay attention to such problems, we should not abuse them. Some students begin to use knowledge of different subjects because of nervousness and sensitivity. In this way, they will be deducted a lot of points. This is not right. In the face of exams, we should try to relax, first think about ideas, how to solve obstacles, and then use them when we find that we can solve them with the knowledge of other disciplines to ensure that we don't lose points.
(4) Now the middle angle operation in mathematics has a cross-scientific trend, which is the result of the development of knowledge. I believe there will be more updated comprehensive questions under this trend. I only hope that we can follow the trend and make progress together!
8 June 2007 10