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Mathematical Modelling in Practice ⇐ ПредыдущаяСтр 5 из 5
Mathematics can be used to " model", or represent, how the real world works. Example: how much space is inside this cardboard box?
So we have a very simple mathematical model of how much space that box contains. Accurate? The model is not the same as the real thing. In our example we did not think about the thickness of the cardboard, or many other " real world" things. But hopefully it is good enough to be useful.
But maybe you need more accuracy, maybe you send hundreds of boxes every day, and the thickness of the cardboard worries you. So let's see if we can improve the model: The cardboard is " t" thick, and all measurements are outside the box... how much space is inside?
Now we have a better model. Still not perfect (did we consider wasted space because we could not pack things neatly, etc...), but better. So a model is not reality, but should be good enough to be useful. Playing With The Model Now we have a model, we can use it in different ways: Example: Your company uses 200x300x400 mm size boxes, and the cardboard is 5mm thick. Someone suggests using 4mm cardboard... how much better is that? Let us compare the two volumes: · Current Volume = (200-2× 5) × (300-2× 5) × (400-2× 5) = 21.489.000 mm3 · New Volume = (200-2× 4) × (300-2× 4) × (400-2× 4) = 21.977.088 mm3 That is a change of: (21.977.088-21.489.000)/21.489.000 ≈ 2% more volume So the model is useful. It lets you know you will get 2% more space inside the box (for the same outside measurements). But there are still " real world" things to think about, such as " will it be strong enough? " Thinking Clearly To set up a mathematical model you also need to think clearly about the facts! Example: on our street there are twice as many dogs as cats. How do you write this as an equation? · Let D = number of dogs · Let C = number of cats Now... is that: 2D = C or should it be: D = 2C
(2D = C is a common mistake, as the question is written " twice... dogs... cats" ) Here is another one: Example: You are the supervisor of 8-hour shift workers. They recently had their break times reduced by 10 minutes but total production did not improve. At first glance there is nothing to model, because there was no change in production. But wait a minute... they are working 10 minutes more, but producing the same amount, so production per hour must have dropped! Let us assume they used to work 7 hours (420 minutes): Change in production per hour = 410/420 = 0, 976... Which is a reduction of more than 2% But even worse: the first few hours of the shift would not be affected by the shorter break time, so it could be a 4 or 5% reduction later in the shift. You could recommend: · looking at production rates for every hour of the shift · trying different break times to see how they affect production
A Bigger Example: Most Economical Size OK, let us have a go at building and using a mathematical model to solve a real world question.
Your company is going to make its own boxes!
It has been decided the box should hold 0, 02m3 (0, 02 cubic meters which is equal to 20 liters) of nuts and bolts. The box should have a square base, and double thickness top and bottom. Cardboard costs $ 0, 30 per square meter. It is up to you to decide the most economical size.
Step One: Draw a sketch! It helps to sketch out what you are trying to solve!
Step Two: Make Formulas. Ignoring thickness for this model: Volume = w × w × h = w2h And we are told that the volume should be 0, 02m3: w2h = 0, 02 Areas: Area of the 4 Sides = 4 × w × h = 4wh Area of Double Tops and Bases = 4 × w × w = 4w2 Total cardboard needed: Area of Cardboard = 4wh + 4w2
Step Three: Make a Single Formula For Cost We want a single formula for cost: Cost = $0, 30 × Area of Cardboard = $0, 30 × (4wh + 4w2) And that is the cost when we know width and height. That could be hard to work with... a function with two variables. But we can make it simpler! Because width and height are already related by the volume: Volume = w2h = 0, 02 ... which can be rearranged to... h = 0, 02/w2 ... and that can be put into the cost formula... Cost= $0, 30 × (4w× 0, 02/w2 + 4w2) And now the cost is related directly to width only. With a little simplification we get: Cost= $0, 30 × (0, 08/w+ 4w2)
Step Four: Plot it and find minimum cost What to plot? Well, the formula only makes sense for widths greater than zero, and I also found that for widths above 0, 5 the cost just gets bigger and bigger. So here is a plot of that cost formula for widths between 0, 0 m and 0, 55 m: x is width, and y is cost Just by eye, I see the cost reaches a minimum at about (0, 22, 0, 17). In other words: · when the width is about 0, 22 m (x-value), · the minimum cost is about $0, 17 per box (y-value). In fact, looking at the graph, the width could be anywhere between 0, 20 and 0, 24 without affecting the minimum cost very much. Step Five: Recommendations Using this mathematical model you can now recommend: · Width = 0, 22 m · Height = 0, 02/w2 = 0, 02/0, 222 = 0, 413 m · Cost = $0, 30 × (0, 08/w+ 4w2) = $0, 30 × (0, 08/0, 22+ 4× 0, 222) = $0, 167 Or about 16, 7 cents per box But any width between 0, 20 m and 0, 24 m would be fine. You might also like to suggest improvements to this model: · Include cost of glue/staples and assembly · Include wastage when cutting box shape from cardboard. · Is this box a good shape for packing, handling and storing? · Any other ideas you may have! Predicting the Future Mathematical models can also be used to forecast future behavior. Example: An ice cream company keeps track of how many ice creams get sold on different days.
Computer Modeling Mathematical models can get very complex, and so the mathematical rules are often written into computer programs, to make a computer model.
Fractions A fraction is a part of a whole. If we divide a cake into five equal-sized pieces, we are dividing it into fifths. One piece of our cake is a fifth of the whole. This is a fraction, and mathematicians would write this fraction as 1/5. This way of writing down fractions tells us two important things. The number at the bottom of the fraction tells us into how many parts the cake (the whole) has been divided. In this case, the cake has been divided into fifths. The number at the top of the fraction tells us how many of the slices of the whole we have.
Curves Much of our knowledge of geometric curves comes from the work of the Greek mathematician Hypatia (AD 370-415).She developed the studies of the Greek mathematician Apollonius (262 BC-190 BC). Hypatia formed common curves, such as the circle, the ellipse, the parabola, and the hyperbola, by slicing a cone at different angles. This method of creating curves is known as conic sectioning. Another way of creating a geometric curve is to trace the path of a point as it moves according to certain conditions. For example, by tracing the path of a moving point that is always the same distance from another fixed point, we can form a circle.
заключение
Учебно-методическое пособие по английскому языку, ориентированное на студентов первого года обучения факультета «Математика и Естественные Науки» профиля 010800 «Механика и Математическое моделирование», способствует формированию различных навыков. Упражнения на развитие языковых и речевых навыков в пособии нацелены на формирование профессионально-ориентированного вокабуляра и основ коммуникативной компетентности. Пособие предназначено не только для становления и совершенствования специальных лингвистических и коммуникативных компетентностей, но и формирования и развития учебно-познавательных компетентностей, предполагающих самостоятельную творческую и поисковую работу студентов. С этой целью в пособие включены упражнения под рубрикой «Проектная работа» (Project Work), требующая поисковых и презентативных навыков обучаемых. Библиография: 1. Ellen, S. http: //www.ehow.com/list_6370184_math-requirements-diesel-mechanics.html (accessed March 15, 2014) 2. Hewitt, D. Proceedings of the British Society for Research into Learning Mathematics 26 (2) June 2006. – P. 92 – 93. (http: //www.bsrlm.org.uk/IPs/ip26-2/BSRLM-IP-26-2-16.pdf.(accessed April 6, 2014)) 3. Степанова, О.В. Topics. Dialogues. Discussions. – Ижевск: Изд-во ИжГТУ, 2010. – 128 с. 4. http: //www.ehow.com/how-does_5008919_mechanics-use-math-their-job.html (accessed March 10, 2014) 5. http: //www.ehow.com/list_6370184_math-requirements-diesel-mechanics.html (accessed April 28, 2014) 6. http: //www.mathsisfun.com/algebra/mathematical-models.html (accessed March 16, 2014) 7. Modelling and Mechanics // ULR: http: //www.cimt.plymouth.ac.uk/projects/mepres/alevel/mechanics_ch1.pdf (accessed March 11, 2014) 8. http: //www.youtube.com/watch? v=c2bO20DFcpA (accessed April 5, 2014) 9. http: //www.youtube.com/watch? v=GnlgGmLNn5o (accessed May 5, 2014) 10. http: //www.youtube.com/watch? v=i4fhMtDhAFQ (accessed April 9, 2014) 11. http: //www.youtube.com/watch? v=i6VBPiA-eBk (accessed April 18, 2014) 12. http: //www.youtube.com/watch? v=QG6zpNL-vek (accessed May 2, 2014) 13. http: //www.youtube.com/watch? v=Rkzxr1ojA4I (accessed May 3, 2014)
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