Mathematical Modelling in Practice

Mathematics can be used to " model", or represent, how the real world works.

Example: how much space is inside this cardboard box?

We know three measurements: · h (height), · w (width), and · l (length), and (luckily! ) the formula for the volume of a cuboid is: Volume = h × w × l

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.

If you get charged by the volume of the box you are sending, you can take a few measurements and know how much to pay. It can also be useful when deciding which box to buy when you need to pack things. So the model is 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?

The inside measurements need to be reduced by the thickness of each side: · The inside height will be h-2t · The inside width will be w-2t, · The inside length will be l-2t and now the formula is: Inside Volume = (h-2t) × (w-2t) × (l-2t)

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!

The base is square, so we will just use " w" for both lengths

The box has 4 sides, and double tops and bottoms. The box shape could be cut out like this (but would probably be more complicated):

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:

Plot of y= 0, 3(0, 08/x+4x²)

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.

By comparing this to the weather on each day they can make a mathematical model of sales versus weather. They can then predict future sales based on the weather forecast, and decide how many ice creams they need to make... ahead of time!

Computer Modeling

Mathematical models can get very complex, and so the mathematical rules are often written into computer programs, to make a computer model.

Appendix AUDIOSCRIPTS: Language of Math Mathematics has a language of its own, which uses numbers and symbols instead of words and punctuation. The earliest recorded numbers were marks made on a stick. These marks were made in small groups of, for example, two or five. Eventually these groups were given symbols of their own (2, 5, etc.) and a system of arithmetic developed. Mathematicians introduced special symbols to replace words such as " plus" and " eguals". They also introduced special words to express new ideas. Terms such as " triangle" and " sguare", for example, were applied to figures that are geometrically defined. Powers Calculations that involve the repeated multiplication of a number can be written down in a simpler form. An easier way of writing 2x2x2x2 is to place a small 4 above the number 2. This indicates the number of twos that are to be multiplied together. Mathematicians would say that we have multiplied the number 2 to the 4th power. Take care to remember that the two the 4th power is not the same thing as 2 x 4. Charts Like Graphs, charts display numerical information in a way that is easy to understand, using pictures or diagrams to take the place of numbers. Bar charts substitute bars for data. The length or height of each bar show a quantity. The longer or higher the bar, the greater the quantity. Pie charts display information by dividing a circle into portions. Each portion of the circle represents a percentage of the total. Pictograms use images to present information. Different-sized images are used to represent different quantities. The larger the image, the larger the quantity. Sequences A mathematical sequence follows a particular rule. For example, 2, 4, 6, 8 is a sequence of even numbers. Any sequence of numbers that increase by the same amount each time is called an arithmetic sequence. Sequences often occur in nature. For example, when single-celled organisms reproduce by splitting into two parts, then four, and so on, they are following a pattern known as a geometric sequence и т.д., (1, 2, 4, 8, 16, 32, etc.).

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)

⇐ Предыдущая 1 2 3 45