2023年12月20日发(作者：国华图书数学试卷)
Winter is approaching, may the dragon’s wings grow more
abundant
Summary
In the game of thrones, Daenerys Targaryen depicts the image of a dragon. In eastern and
western cultures, the phenomenon of dragons is not uncommon. If dragons live in modern society,
how can we raise these war monsters? Research, and applied the cross disciplines of biology, physics,
and chemistry to build a mathematical model and solve it to achieve the maximum growth of the
dragon. Of course, dragons do not exist in real life, so we likened pterosaurs, modern Aircraft and
chemical burner to derive the specific physiological characteristics of the dragon to ensure the
rationality and scientificity of the research.
First, we studied the flight and fire-spitting models of dragons. Through analogical reasoning,
our hypothetical dragon\'s fire-spitting principle is similar to modern alcohol flamethrowers. For
dragon flight, we used fluid mechanics to get the dragon\'s flight speed. And glucose energy loss.
Combining the two to get the energy loss model of the dragon. Second, we studied the basic physical
characteristics of the dragon. For the relationship between the body length and body age of the
dragon, we established an elastic model of growth. Because the weight and body length of dragons
have upper and lower limits, in order to comply with basic ecology, we have defined the dragon\'s
bone saturation value as the cut-off value, and conducted a segmented study. When studying the
relationship between weight and body length, We know that the weight of the dragon is proportional
to the cube of the body length. Then, because the dragon needs resources to replenish like other
animals, we built a dragon\'s food supply model. Suppose that the three dragons have the same
competitiveness and the daily sheep Resources are the same. According to ecology, when the
number of sheep in a certain area reaches k / 2, we need to migrate the dragon. Finally, the
temperature will affect the living environment of the dragon, so the dragon needs to followMigration
was selected for changes in temperature, and we selected three areas of drought, cold, and warmth
to study the dragon, and integrated the model of the regional area of the dragon by the appealing
model.
In addition, we wrote a letter to the author of the Song of Ice and Fire, giving some suggestions
on the actual ecological foundation of the dragon, hoping to be adopted. Although the dragon does
not exist in our real life, the dragon can be broken down into Part of our modern society. For the
dragon\'s flying spitfire energy loss model, we can further study the aircraft\'s fluid mechanics and
modern flamethrowers. The study of non-existent organisms also prepares us for the arrival of new
species .
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table of Contents
Winter is approaching, may the dragon’s wings grow more abundant ............................................. 1
Summary ............................................................................................................................... 1
table of Contents .................................................................................................................... 2
1 Introduction ........................................................................................................................ 3
1.1 restatement ................................................................................................................... 3
1.2 3
2 Assumptions and reasons ................................................................................................... 4
3 Symbol Definition .............................................................................................................. 4
4. Mathematical modeling ..................................................................................................... 5
4.1 About Dragon Flight and Spitfire Consumption ......................................................... 5
4.2 About the relationship between dragon\'s body length and weight and age ................. 7
4.3 About Dragon\'s Food Supply ...................................................................................... 8
4.4 Regulating the area of dragons by region .................................................................... 9
5 Sensitivity analysis ........................................................................................................... 10
6 Model evaluation and outlook .......................................................................................... 11
6.1 Model evaluation ....................................................................................................... 11
6.2 Further discussion ...................................................................................................... 12
7 to a letter from George RR Martin ................................................................................... 12
ix: ......................................................................................................................... 13
8.1 References ................................................................................................................. 13
8.2 Matlab code ............................................................................................................... 13
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1 Introduction
1.1 restatement
In the magical TV series \"Game of Thrones\", Daenerys Targaryen, known as the Mother of
Dragons, raised three dragons as an aggressive army. Dragons have always been the most mysterious
monsters in Eastern and Western cultures, but if Dragons live in the present era, how should we feed
the three dragons in pursuit of maximum growth? In this article, we assume that the growth rules of
dragons are in line with basic biology. To study them, we build mathematical models to solve
problem.
a. Analyze the change of the dragon\'s weight length with age, and estimate the value of the
dragon\'s weight length corresponding to the age group.
b. Investigate the loss of self energy during dragon fire, flight, and breathing, so as to estimate
the minimum supply value of dragon for external activities
c. Dragons need food and survival areas like other animals in the real world. Through certain
assumptions and calculations, we can determine the total amount of food that dragons need daily
and the size of living areas in three areas.
d. Sensitivity analysis: As temperature and climate change, dragons will also migrate to different
regions. Therefore, we need to analyze the differences in the impact of dragons on the survival of
arid regions, temperate regions, and cold regions.
1.2 Problem Analysis
Because dragons do not exist in real life, we need to use some things in the real world to compare
dragons in order to achieve the purpose of studying dragons. In analyzing the biological
morphological characteristics of dragons, we use the knowledge of ecology and basic elements of
biology Let\'s conceive the basic biological characteristics of the dragon such as weight and body
length. For the energy loss model of the dragon, we have studied three aspects to describe its loss.
Here we compare the modern flamethrower and establish related chemical equations to achieve the
research of the dragon. Spitfire loss. In addition, in TV series such as \"Game of Thrones\" we will
find that dragons can fly in common sense, so we have derived the dragon\'s flight loss. Of course,
all aerobic organisms can breathe. Dragons are no exception, so there is a loss of breathing to
maintain body temperature. At the same time, in order to make up for the loss of dragons in daily
activities, we have established a material reserve model, in which materials are cattle and sheep in
real life, etc. Finally, during the cyclical changes in climate and food, the dragons we feed will also
migrate to some extent, so we analyzed the impact of different regions on the growth of
account various factors that we can more scientific training of dragons, have achieved our purpose.
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2 Assumptions and reasons
After a comprehensive analysis of the problem, in order to increase the enforceability, we make
the following assumptions to ensure the rationality of our model establishment.
2.1 Assumptions: The basic biological characteristics of dragons are in line with the law of
biological growth. In modern life, the growth and development of dragons should also be similar to
other animals and conform to basic biology.
2.2 Assumption: The dragon will spit fire and fly, and its flight conforms to the physical environment
of fluid mechanics
Reason: In Game of Thrones, the image of the dragon was once able to fly and spit fire.
2.3 Assumption: In the single field we are studying, the environment of a certain area will not change
abruptly and maintain a dynamic stability.
2.4 Hypothesis: Dragons are top predators in the food chain, but dragons do not cause devastating
harm to the biosphere.
2.5 Assumption: The weight distribution of the dragon is uniform, and the body length reaches 30
to 40 cm at the time of birth.
Reason 2.6: We refer to ancient biology and some dinosaur fossils.
2.7 Hypothesis: Except for the skull, heart, liver, lungs, kidneys, bones, etc., the sum of other body
masses is proportional to the cube of height.
Reason: The hypothesis is obtained by counting the relationship between body length and weight of
modern organisms.
2.8 Hypothesis: The dragon is a constant temperature animal whose body temperature is not affected
by external factors.
Reason: A few pterosaur fossils have traces of \"hair\" on the surface, while the dragons in Game
of Thrones are similar to pterosaurs.
2.9 Hypothesis: The dragon is fully aerobic during the flight to provide energy
2.10 Hypothesis: A certain fixed ratio of the amount of energy that is not assimilated by the growth
and metabolism of the dragon\'s breathing and other organisms
2.11 Hypothesis: Dragon\'s Flight Similar to Modern Fighter
3 Symbol Definition
symbol
M
L
a%
b%
description
quality
Body length
Unassimilated amount is a fixed ratio of intake
Respiratory and other biological growth
metabolic consumption can be a fixed ratio of
assimilation amount
time
Acceleration
Consume energy
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t、T、i
A
W
N
d%
K
e%
Q
S
S
R
V
Y
Amount of cattle and sheep
Assimilation rate of cattle and sheep grazing in
the area
Environmental capacity
Solar utilization
Solar energy per unit area
Flying distance of dragon
area
Parameters of environmental carrying capacity
Speed or rate
age
4. Mathematical modeling
4.1 About Dragon Flight and Spitfire Consumption
4.1.1 Proposed model
Considering that dragons fly and spit fire during activities, we have established an energy loss
model. Comparing the principle of dragon\'s spitfire with modern flamethrowers, modern
flamethrowers consume hydrocarbons or alcohols. It does not cause any impact, so the dragon\'s fire-breathing principle is in line with the alcohol flame-thrower principle. Considering that the
formaldehyde produced by the metabolism of methanol in the animal body is harmful to the body,
we stipulate that ethanol is the fuel used by the dragon\'s flame. In the process, the relationship
between the dragon\'s flight speed and glucose energy consumption is obtained according to fluid
mechanics. In this process, we assume that the aerobic respiration is completely performed, and the
energy consumed by the dragon due to flight is obtained according to the glucose consumption. In
summary, the dragon energy loss model is obtained. .
4.1.2 Establishment and Solution of Dragon\'s Spitfire Model
The thermochemical equation for ethanol combustion is: C2H5OH (l) + 3O2 (g) = 2CO2 (g) +
2H2O (l) △ H = -12KJ / g
Specify the energy released per unit mass of ethanol combustion x1
When the dragon spit fire in unit time t, the unit mass of ethanol consumption is a fixed value
The energy consumed by the fire time t1 is w1
The mass consumed by the fire time T1 is m4
Let the energy emitted by the combustion of unit mass of ethanol be w1 \'
Then W1 = x1 * t
m4=W1/W1’
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Solve m4 = x1 * t / W1 \'
4.1.3 Establishment and Solution of Dragon Flight Model
During the flight of the dragon, it will be affected by the air resistance. In the ideal situation, the
dragon\'s flight can be considered as a uniform acceleration and then a uniform speed, and it will
decelerate when it is about to reach its destination.
When Long uniform acceleration is specified, the acceleration is a
Since the flight of the dragon is similar to that of a fighter, a = 30m / s ^ 2
The speed of the dragon during uniform motion is v0
The total flight length of the dragon during flight is s
Because air resistance is proportional to the speed of movement, that is, F1 = k * v (where k is a
constant)
Since the dragon\'s flight is similar to an airplane, we can get k = 3.2325
Available according to the relevant kinematic formula
The flying distance of the dragon during uniform acceleration is s1 = (v0) ^ 2 / 2a
The flying distance of the dragon during uniform deceleration is s3 = (v0) ^ 2 / 2a
The flying distance of the dragon during uniform motion is s2 = s-s1-s3
Average air resistance during uniform acceleration F1 \'= k * (0 + v0) / 2
The average air resistance during uniform motion is F1 \'\' = k * v0
Average air resistance during uniform deceleration f1 \'\' \'= k * (v0 + 0) / 2
According to the law of conservation of energy
The energy w2 consumed by the dragon during flight is all used for air resistance work
W2=F1’*s1+F1’’*s2+F1’’’*s3
Solve W2 = 3.2325 * v0 * s-3.2325 * (v0) ^ 3 / (2 * 30)
During the flight of the dragon, the principle of energy provided by aerobic respiration is
C6H12O6+6O2=6CO2+6H2O
Among them, the energy produced when 1g of glucose is completely consumed is 16KJ
Then the weight consumed in this process is m6 = W2 / 16
[v,s]=meshgrid(0:0.1:100;0:0.1:100);
m=3.2325*v*s-3.2325*v^3/60
mesh(v,s,m)
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4.2 About the relationship between dragon\'s body length and
weight and age
4.2.1 Proposed Model
First, in order to study the relationship between the weight, length, and age of the dragon, that is,
morphological characteristics, we established a model of elasticity during growth. The above-mentioned change curve is continuous, so we use the weight of the dragon at birth, and consider the
weight and length of the dragon. The relationship between age changes can be used to derive the
normal weight and body length of dragons in all ages. When analyzing the weight changes of
dragons, biological knowledge shows that the amount of assimilation of the dragon is equal to the
intake amount minus the amount of unassimilated amount Considering that the growth rate of the
dragon in adulthood is a watershed, we use the saturation value of the dragon\'s head, heart, and liver
as a cutoff value to estimate the relationship between the dragon\'s weight and age, respectively.
When studying the body length of the dragon, according to the existing morphological knowledge,
the head to hip of the dragon is used as the length standard. Because the weight of the dragon is
proportional to the cube of the dragon\'s length, we get the weight and length Functional relationship.
Of course, the daily weight gain of the dragon must be less than the daily energy consumption. In
summary, we have a dragon intake model.
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4.2.2 Model establishment
Specify the weight of the dragon as m
Dragon was born with a weight of m0 (known m0 = 10kg)
Assume that the mass of cattle and sheep fed by a train every day is m2
The assimilation amount of the dragon is fixed at a%
A certain fixed ratio of the amount of unabsorbed energy due to growth and metabolism of
organisms such as dragon\'s respiration, recorded as b%
The weight gain of the dragon is m \'
The sum of the weight of the dragon\'s head, heart, liver, lungs, kidneys, bones, etc. m1 increases
with age y until adulthood
Dragon is y1 when he is an adult
The growth rate of m1 is v1
The mass of m1 at birth is m0
Before the dragon reaches y1
m1=m0+v1*y
After the dragon reaches y1
m1’=m0+v1*y1
4.2.3 Model Solving
m’=m2*(1-a%)*(1-b%)-m4-m6
So the weight of the dragon m = m \'+ m0
Except for the dragon, except for the head, heart, liver, lungs, kidneys, bones, etc., the sum of
other body masses is proportional to the cube of height, and the body length is recorded as l
When the age of the dragon does not reach y1, l = (m-m1) ^ (1/3)
When the age of the dragon reaches y1, l \'= (m-m1\') ^ (1/3)
M2 =
y=0:0.1:20
function[y]= (m2*(1-a%)*(1-b%)-m4-m6-v1*y)
y=20:0.1:100
function[y]= (m2*(1-a%)*(1-b%)-m4-m6-v1*20)
power(y,1/3)
4.3 About Dragon\'s Food Supply
4.3.1 Proposed model
Based on the above analysis, we studied the living area of the three dragons in the region and
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their impact on the ecological community in the region. For the sake of research, we assume that
the other creatures in the region are cattle and sheep, and the competitiveness of the three dragons
is comparable, Being a top predator in the food chain.
4.3.2 Model establishment
The local food chain can be approximated as: grass → cow or sheep → dragon
Assume that the weight of the grass in the arid region, the warm temperate region, and the Arctic
region is the same as m8.
Remember that the mass of each cow and sheep is the same as m7
We provide the same initial number of cattle and sheep in all three regions
Assume that the daily growth rate of cattle and sheep is c%
The initial number of cattle and sheep is n1
And n1 is the number of populations reaching k in the region
Dragons live in this area. When the number of cattle and sheep reaches k / 2, in order to ensure the
balance of the ecological environment, the dragons need to be moved to other regions.
4.3.3 Model Solving
The initial amount of cattle and sheep on day 1 is: n1
The initial amount of cattle and sheep on the second day is: N2 = (N1-3 * m2 / m7) * (1 + c%)
The initial amount of cattle and sheep on the third day is: N3 = ((N1-3 * m2 / m7) * (1 + c%)-3 *
m2 / m7) * (1 + c%)
……
From this we can get the initial amount of Ni of cattle and sheep on day i
I can be solved by the equation Ni = K / 2
That is, the dragon needs to change a living area after living in the area for i days.
4.4 Regulating the area of dragons by region
4.4.1 Proposed model
In order to ensure the normal growth of the dragon, we provide fixed-quality cattle and sheep as
the supply of resources for the survival of the dragon region, and assume that the number of cattle
and sheep is proportional to the size of the regional living area. Considering the growth rate of cattle
and sheep, we have established a differential The equation draws the relationship between the
growth rate of cattle and sheep and the age of the dragon. However, cattle and sheep will reach a
growth saturation value at a certain moment, we will consider it in segments to ensure that the data
is more scientific. In order to comply with ecology, cattle The supply of sheep should also have a
lower limit. In summary, we have established a dragon-cow-sheep-living area function model.
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4.4.2 Model establishment
Remember that the assimilation rate of cattle and sheep grazing in this area is d%
Because the solar energy received by the surface area of the three areas is different, the total area
required for the grass under the same quality conditions is different. The utilization rate of the solar
energy is required to be e% (0.5 The solar energy per unit area in the arid area is q1 Unit area solar energy in warm zone is q2 Solar energy per unit area in the Arctic is q3 4.4.3 Model Solving According to the utilization of solar energy, we can find: Area required to support the arid areas where the three dragons live: S1 = m8 / (q1 * e%) Warm zone: S2 = m8 / (q2 * e%) Arctic region: S3 = m8 / (q3 * e%) 5 Sensitivity analysis Impact of climatic conditions on dragon life The effect of climatic conditions on dragon growth can be obtained from the logistic growth model dm/dt=r*m*(1-m/k) That is m = 15 / (4 * t + 20); (Where m is the mass that the dragon can eventually grow into) Where m0 = 10 (k is the maximum carrying capacity of the ecosystem and r is a parameter of the environmental carrying capacity) k is 0.75 r is 0.8 dm/dt=0.8*m*(1-m/0.75) t=0:0.1:100; m=15./(4*t+20); plot(t,m) 10 / 13 6 Model evaluation and outlook 6.1 Model evaluation For the idealized model of Yanglong, we have performed various aspects of modeling and solving, and the scope is relatively broad. Of course, the content has been streamlined to facilitate understanding and application. We have used physical and biological models based on The mathematical formulas are also encountered in the middle school stage. In these more basic models, we have solved efficiently, and at the same time, for the interdisciplinary problems of question a, we have considered the field that the ideal biology of dragons may involve and solve The process is relatively complete. In addition, the four models are closely related and logical. First, we consider the consumption of dragons in daily life, and use the results of consumption to calculate the weight and length of the dragon at various ages. In order to meet the requirements of all ages, we have established the ecological supply model of dragons, and discussed the problem of periodic alternating fields. Second, the fields are also scoped. Therefore, we calculated the scope of three areas with different climates. Interval problems. However, the models we build are idealized, the data is also streamlined, and the assumptions set are also fallible. In reality,The data is diverse and complex, and our considerations are obviously lacking, and further optimization is needed in the later stage. In summary, the model we built is very consistent with the solution of the problem. Although there are some flaws, it does not affect the specific Specific analysis of the problem. 11 / 13 6.2 Further discussion Combining the models and evaluations described above, we will improve in the later stages. If this model is used in a specific environment, by statistic large amounts of real data, we can optimize the model. At the same time research also It will be more scientific and rigorous, and it will be more efficient for raising a fictional creature. 7 to a letter from George RR Martin Dear George RR Martin Hope you are well After reading the Song of Ice and Fire, we watched the \"Game of Thrones\". We became very curious about the mysterious giant that appeared in it-the dragon. Dragons are not uncommon in Eastern and Western cultures. In previous impressions However, there are few studies on dragons. So if we imagine that dragons live in modern times, what would it look like? According to the description of the dragon in the novel, we discussed the following questions. What are the ecological impacts and requirements of the dragon? What is the energy consumption of the dragon, what are their calorie intake requirements? How much area is needed to support the three dragons? Energy loss during fire? In response to these problems, we constructed a multivariate non-linear objective programming model of dragon\'s growth index and function, size, diet, growth changes, and other animal-related features. Considering the physical characteristics of dragons, we will Its fire-spitting ability is analogized to modern flame-throwers to ensure scientific and rational research. Based on these, we have established a mathematical model. The weight and length of the dragon also grows with the age of the dragon. When the dragon grows slowly at the initial 10 kilograms, the mass of sheep it needs each year also varies The growth of the supply chain of resources and the size of the ecological community should also change. The fire and flight of the dragon will also have a certain impact on the ecological environment. As the dragon and other creatures will migrate with changes in temperature, we choose The three regions of the cold zone, temperate zone and arid zone were taken as key research objects to find out the impact of climate change on Long. Therefore, we make the following suggestions, hoping that the survival of the dragon in the realm of science is more reasonable and scientific. When the herd resource is saturated, the dragon needs to expand the area living area. Dragons like warm, hydrated areas, and migrate to warm areas in the cold winter. A dragon has a certain weight and length when it is just born, and it will grow over time, but it also has an upper limit. It cannot grow endlessly. The daily energy intake of the dragon is limited, and the dragon spitfire flight consumes energy, which requires that the dragon\'s flight distance and spitfire time are limited, and it is related to the age of the dragon body. Because the living conditions of the three areas are different, the unit area will also receive solar energy differently, resulting in different resource distributions in each area, which means that 12 / 13 the speed of dragon growth should also be different in different areas. The environmental carrying capacity of each area is limited, and the dragon does not stay in one place for long. The above content is the result of our research on the Queen of Dragons. We sincerely hope that you can adopt it, and we have been looking forward to your new book. Your fans: 27 groups January 7, 2020 ix: 8.1 References 1) Chen ch on Environmental Carrying Capacity of Yuhuan County [j] .Energy and Energy Conservation, 2014 (4): 31-33. 2) Zhu namic design of modern aircraft [m] .Beijing: National Defense Industry Press, 2011-10-1 3) Jin nmental Ecology [m] .Higher Education Press, 1992 8.2 Matlab code Modeling the flight of a dragon [v,s]=meshgrid(0:0.1:100;0:0.1:100); m=3.2325*v*s-3.2325*v^3/60 mesh(v,s,m) Sensitivity Analysis of the Impact of Climate Conditions on Life t=0:0.1:100; m=15./(4*t+20); plot(t,m) 13 / 13
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