A mathematical model is a way of describing a real life situation (task) using a mathematical language. Real situation Mathematical model Christina and Gleb have the same number of stamps x = y Christina has 6 more stamps than Gleb x + 6 = y x - 6 = y x + y= 6 Gleb has 4 times more stamps than Christina 4x = y x = y. 4y:x=4

The first worker completes the task in t hours, and the second one completes the same task in v hours, while the first worker works 3 hours more than the second.

Three kilograms of apples cost the same as two kilograms of pears. At the same time, it is known that 1 kg of apples costs x r., and 1 kg of pears costs x r. X r. at the river

The cost of a glass of tangerine juice is a p., and a glass of grape juice is b p. It is known that 5 glasses of grape juice cost the same as 6 glasses of tangerine juice.

A cyclist with a speed v 1 and a motorcyclist with a speed v 2 left points A and B at the same time towards each other and met after t hours.

A car with speed v 1 and a bus with speed v 2 v1v1 v2v2 left point A simultaneously in opposite directions A Movement in opposite directions v = v 1 + v 2

From point A, a car and a truck left simultaneously in the same direction, the speeds of which are x km/h and y km/h, respectively. X km/h Y km/ht Movement in one direction v = x-y

A cyclist left point A. At the same time, from point B, 30 km away in the direction of the cyclist, a pedestrian left in the same direction at a speed of x km/h. It is known that the cyclist caught up with the pedestrian after t h. 30 kmt x km/h

12 In the course of solving problems in an algebraic way, reasoning is divided into three stages: drawing up a mathematical drawing up a mathematical model; models; work with mathematical work with a mathematical model (solution of an equation) model (solution of an equation) answer to a question of a problem. answer to the question of the task. Stages of mathematical modeling

Most of life's problems are solved as algebraic equations: by reducing them to the simplest form, i.e. to the compilation of a unified mathematical model. The method of introducing a new variable allows, when solving trigonometric, exponential, logarithmic equations and inequalities, to move on to compiling a single, simpler model: a quadratic equation or inequality.

Example 1. Solve Equation 4 x + 2 x + 1 - 24 = 0.

Decision.

1. First stage. Drawing up a mathematical model.

Noticing that 4 x \u003d (2 2) x \u003d 2 2x \u003d (2 x) 2, and 2 x + 1 \u003d 2 2 x , we rewrite the given equation in the form (2 x) 2 + 2 2 x - 24 = 0.

It makes sense to introduce a new variable: y = 2 X ; then the equation will take the form 2 + 2y - 24 = 0. The mathematical model has been compiled. This is a quadratic equation. 2. Second stage. Working with the compiled model. By solving the quadratic equation 2 + 2y - 24 = 0 with respect to y, we find: y 1 = 4, y 2 = -6.

3. The third stage. The answer to the problem question.

Since y = 2 x , So we have to solve two equations: 2 x = 4; 2 x = -6.

From the first equation we find: x = 2; the second equation has no roots, since for any values ​​of x the inequality 2 x > 0.

Answer: 2.

Example 2. The problem of finding the largest and smallest values ​​of quantities.

The tank, which looks like a rectangular parallelepiped with a square base, should hold 500 liters of water. At which side of the base will the surface area of ​​the tank (without the lid) be the smallest?

Decision. First stage. Drawing up a mathematical model.

1) Optimized value (O.V.) - tank surface area, since the problem requires finding out when this area will be the smallest. Let's designate O. V. with the letter S.

2) The surface area depends on the measurements of the cuboid. We declare the side of the square that serves as the base of the tank as an independent variable (N.P.); Let's denote it as x. It is clear that x > 0. There are no other restrictions, so 0

3) If the tank holds 500 liters of water, then the volume V of the tank is 500 dm 3 . If h is the height of the tank, then V = x 2 h, whence we find h=The surface of the tank consists of a square with side x and four rectangles with sides x and. Means,

S \u003d x 2 + 4 x \u003d x 2 +.

So, S = X 2 + , where x € (0; + ) (we took into account that V = 500)

The mathematical model of the problem has been compiled.

Second phase. Working with the compiled model.

At this stage, for the function S = x 2 + , where x € (0; + )

You need to find a / hiring. This requires the derivative of the function:

S" \u003d 2x -;

S" = .

There are no critical points on the interval (0; + oo), and there is only one stationary point: S" = 0 for x = 10.

Note that for x 10, the inequality S "> 0 is satisfied. Hence, x \u003d 10 is the only stationary point, and the minimum point of the function on a given interval, and therefore, according to the theorem from paragraph 1, at this point the function reaches its smallest value.

Third stage. The answer to the problem question.

The problem asks what side of the base should be in order for the tank to have the smallest surface. We found out that the side of the square that serves as the base of such a tank is 10 dm.

Answer: 10 dm.

What is a mathematical model?

The concept of a mathematical model.

A mathematical model is a very simple concept. And very important. It is mathematical models that connect mathematics and real life.

In simple terms, a mathematical model is a mathematical description of any situation. And that's it. The model can be primitive, it can be super complex. What is the situation, what is the model.)

In any (I repeat - in any!) business, where you need to calculate something and calculate - we are engaged in mathematical modeling. Even if we don't know it.)

P \u003d 2 CB + 3 CB

This record will be the mathematical model of the expenses for our purchases. The model does not take into account the color of the packaging, expiration date, politeness of cashiers, etc. That's why she model, not a real purchase. But the costs, ie. what we need- we'll know for sure. If the model is correct, of course.

It is useful to imagine what a mathematical model is, but this is not enough. The most important thing is to be able to build these models.

Compilation (construction) of a mathematical model of the problem.

To compose a mathematical model means to translate the conditions of the problem into a mathematical form. Those. turn words into an equation, formula, inequality, etc. Moreover, turn it so that this mathematics strictly corresponds to the original text. Otherwise, we will end up with a mathematical model of some other problem unknown to us.)

More specifically, you need

There are an infinite number of tasks in the world. Therefore, to offer clear step-by-step instructions for compiling a mathematical model any tasks are impossible.

But there are three main points that you need to pay attention to.

1. In any task there is a text, oddly enough.) This text, as a rule, has explicit, open information. Numbers, values, etc.

2. In any task there is hidden information. This is a text that assumes the presence of additional knowledge in the head. Without them - nothing. In addition, mathematical information is often hidden behind simple words and ... slips past attention.

3. In any task there must be given communication between data. This connection can be given in clear text (something equals something), or it can be hidden behind simple words. But simple and clear facts are often overlooked. And the model is not compiled in any way.

I must say right away that in order to apply these three points, the problem has to be read (and carefully!) several times. The usual thing.

And now - examples.

Let's start with a simple problem:

Petrovich returned from fishing and proudly presented his catch to his family. Upon closer examination, it turned out that 8 fish come from the northern seas, 20% of all fish come from the southern seas, and not a single one from the local river where Petrovich fished. How many fish did Petrovich buy in the Seafood store?

All these words need to be turned into some kind of equation. To do this, I repeat, establish a mathematical relationship between all the data of the problem.

Where to start? First, we will extract all the data from the task. Let's start in order:

Let's focus on the first point.

What is here explicit mathematical information? 8 fish and 20%. Not a lot, but we don't need a lot.)

Let's pay attention to the second point.

Are looking for covert information. She is here. These are the words: "20% of all fish". Here you need to understand what percentages are and how they are calculated. Otherwise, the task cannot be solved. This is exactly the additional information that should be in the head.

There is also here mathematical information that is completely invisible. This is task question: "How many fish did you buy... It's also a number. And without it, no model will be compiled. Therefore, let us denote this number by the letter "X". We do not yet know what x is equal to, but such a designation will be very useful to us. For more information on what to take for x and how to handle it, see the lesson How to solve math problems? Let's write it right away:

x pieces - the total number of fish.

In our problem, southern fish are given as a percentage. We need to translate them into pieces. What for? Then what's in any the task of the model should be in the same sizes. Pieces - so everything is in pieces. If we are given, let's say hours and minutes, we translate everything into one thing - either only hours, or only minutes. It doesn't matter what. It is important to all values ​​were the same.

Back to disclosure. Whoever does not know what a percentage is will never reveal, yes ... And who knows, he will immediately say that the percentages here of the total number of fish are given. We don't know this number. Nothing will come of it!

The total number of fish (in pieces!) is not in vain with the letter "X" designated. It will not work to count the southern fish in pieces, but can we write it down? Like this:

0.2 x pieces - the number of fish from the southern seas.

Now we have downloaded all the information from the task. Both explicit and hidden.

Let's pay attention to the third point.

Are looking for mathematical connection between task data. This connection is so simple that many do not notice it... This often happens. Here it is useful to simply write down the collected data in a bunch, and see what's what.

What do we have? There is 8 pieces northern fish, 0.2 x pieces- southern fish and x fish- total. Is it possible to link this data somehow together? Yes Easy! total number of fish equals sum of southern and northern! Well, who would have thought ...) So we write down:

x = 8 + 0.2x

This will be the equation mathematical model of our problem.

Please note that in this problem we are not asked to fold anything! It was we ourselves, out of our heads, who realized that the sum of the southern and northern fish would give us the total number. The thing is so obvious that it slips past attention. But without this evidence, a mathematical model cannot be compiled. Like this.

Now you can apply all the power of mathematics to solve this equation). This is what the mathematical model was designed for. We solve this linear equation and get the answer.

Answer: x=10

Let's make a mathematical model of another problem:

Petrovich was asked: "How much money do you have?" Petrovich wept and answered: “Yes, just a little bit. If I spend half of all the money, and half of the rest, then I will have only one bag of money left ...” How much money does Petrovich have?

Again, we work point by point.

1. We are looking for explicit information. You won't find it right away! Explicit information is one money bag. There are some other halves... Well, we'll sort it out in the second paragraph.

2. We are looking for hidden information. These are halves. What? Not very clear. Looking for more. There is another issue: "How much money does Petrovich have?" Let's denote the amount of money by the letter "X":

X- all the money

And read the problem again. Already knowing that Petrovich X of money. This is where the halves work! We write down:

0.5 x- half of all money.

The remainder will also be half, i.e. 0.5 x. And half of the half can be written like this:

0.5 0.5 x = 0.25x- half of the remainder.

Now all the hidden information is revealed and recorded.

3. We are looking for a connection between the recorded data. Here you can simply read the sufferings of Petrovich and write them down mathematically):

If I spend half of all the money...

Let's write down this process. All money - X. Half - 0.5 x. To spend is to take away. The phrase becomes:

x - 0.5 x

and half of the rest...

Subtract another half of the remainder:

x - 0.5 x - 0.25 x

then only one bag of money will remain with me ...

And there is equality! After all the subtractions, one bag of money remains:

x - 0.5 x - 0.25x \u003d 1

Here it is, the mathematical model! This is again a linear equation, we solve, we get:

Question for consideration. Four is what? Ruble, dollar, yuan? And in what units do we have money in the mathematical model? In bags! So four bag Petrovich's money. Good too.)

The tasks are, of course, elementary. This is specifically to capture the essence of drawing up a mathematical model. In some tasks, there may be much more data in which it is easy to get confused. This often happens in the so-called. competency tasks. How to pull mathematical content out of a pile of words and numbers is shown with examples

One more note. In classical school problems (pipes fill the pool, boats are sailing somewhere, etc.), all the data, as a rule, is chosen very carefully. There are two rules:
- there is enough information in the problem to solve it,
- there is no extra information in the task.

This is a hint. If there is some unused value in the mathematical model, think about whether there is an error. If there is not enough data in any way, most likely, not all hidden information has been revealed and recorded.

In competence and other life tasks, these rules are not strictly observed. I don't have a hint. But such problems can also be solved. Unless, of course, practice on the classic.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

First level

Mathematical models at the OGE and the Unified State Examination (2019)

The concept of a mathematical model

Imagine an airplane: wings, fuselage, tail, all this together - a real huge, immense, whole airplane. And you can make a model of an airplane, small, but everything is real, the same wings, etc., but compact. So is the mathematical model. There is a text problem, cumbersome, you can look at it, read it, but not quite understand it, and even more so it is not clear how to solve it. But what if we make a small model of it, a mathematical model, out of a large verbal problem? What does mathematical mean? So, using the rules and laws of mathematical notation, remake the text into a logically correct representation using numbers and arithmetic signs. So, A mathematical model is a representation of a real situation using a mathematical language.

Let's start simple: The number is greater than the number by. We need to write it down without using words, just the language of mathematics. If more by, then it turns out that if we subtract from, then the very difference of these numbers will remain equal. Those. or. Got the gist?

Now it’s more complicated, now there will be a text that you should try to present in the form of a mathematical model, until you read how I will do it, try it yourself! There are four numbers: , and. A product and more products and twice.

What happened?

In the form of a mathematical model, it will look like this:

Those. the product is related to as two to one, but this can be further simplified:

Well, with simple examples, you get the point, I guess. Let's move on to full-fledged tasks in which these mathematical models also need to be solved! Here is the task.

Mathematical model in practice

Task 1

After rain, the water level in the well may rise. The boy measures the time of falling small pebbles into the well and calculates the distance to the water using the formula, where is the distance in meters and is the time of falling in seconds. Before the rain, the time for the fall of the pebbles was s. How much must the water level rise after the rain in order for the measured time to change to s? Express your answer in meters.

Oh God! What formulas, what kind of well, what is happening, what to do? Did I read your mind? Relax, in tasks of this type, conditions are even more terrible, the main thing to remember is that in this task you are interested in formulas and relationships between variables, and what all this means in most cases is not very important. What do you see useful here? I personally see. The principle of solving these problems is as follows: you take all known quantities and substitute them.But sometimes you have to think!

Following my first advice, and substituting all the known ones into the equation, we get:

It was I who substituted the time of the second, and found the height that the stone flew before the rain. And now we need to count after the rain and find the difference!

Now listen to the second advice and think about it, the question specifies "how much the water level must rise after rain in order for the measured time to change by s". You need to figure it out right away, soooo, after the rain the water level rises, which means that the time for the stone to fall to the water level is less, and here the ornate phrase “so that the measured time changes” takes on a specific meaning: the fall time does not increase, but is reduced by the specified seconds. This means that in the case of a throw after the rain, we just need to subtract c from the initial time c, and we get the equation for the height that the stone will fly after the rain:

And finally, in order to find how much the water level should rise after the rain, so that the measured time changes by s, you just need to subtract the second from the first height of the fall!

We get the answer: per meter.

As you can see, there is nothing complicated, most importantly, don’t bother too much where such an incomprehensible and sometimes complex equation came from in the conditions and what everything in it means, take my word for it, most of these equations are taken from physics, and there the jungle is worse than in algebra. It sometimes seems to me that these tasks were invented to intimidate the student at the exam with an abundance of complex formulas and terms, and in most cases they require almost no knowledge. Just carefully read the condition and substitute the known values ​​in the formula!

Here is another problem, no longer in physics, but from the world of economic theory, although knowledge of sciences other than mathematics is again not required here.

Task 2

The dependence of the volume of demand (units per month) for the products of a monopoly enterprise on the price (thousand rubles) is given by the formula

The company's monthly revenue (in thousand rubles) is calculated using the formula. Determine the highest price at which the monthly revenue will be at least a thousand rubles. Give the answer in thousand rubles.

Guess what I'll do now? Yeah, I'll start substituting what we know, but, again, you still have to think a little. Let's go from the end, we need to find at which. So, there is, equal to some, we find what else it is equal to, and it is equal, and we will write it down. As you can see, I don’t particularly bother about the meaning of all these quantities, I just look from the conditions, what is equal to what, that’s what you need to do. Let's return to the task, you already have it, but as you remember, from one equation with two variables, none of them can be found, what to do? Yeah, we still have an unused particle in the condition. Here, there are already two equations and two variables, which means that now both variables can be found - great!

Can you solve such a system?

We solve by substitution, we have already expressed it, which means we will substitute it into the first equation and simplify it.

It turns out here is such a quadratic equation: , we solve, the roots are like this, . In the task, it is required to find the highest price at which all the conditions that we took into account when we compiled the system will be met. Oh, it turns out that was the price. Cool, so we found the prices: and. The highest price, you say? Okay, the largest of them, obviously, we write it in response. Well, is it difficult? I think not, and you don’t need to delve into it too much!

And here's a frightening physics for you, or rather, another problem:

Task 3

To determine the effective temperature of stars, the Stefan–Boltzmann law is used, according to which, where is the radiant power of the star, is a constant, is the surface area of ​​the star, and is the temperature. It is known that the surface area of ​​a certain star is equal, and the power of its radiation is equal to W. Find the temperature of this star in degrees Kelvin.

Where is it clear? Yes, the condition says what is equal to what. Previously, I recommended that all unknowns be immediately substituted, but here it is better to first express the unknown sought. Look how simple everything is: there is a formula and they are known in it, and (this is the Greek letter "sigma". In general, physicists love Greek letters, get used to it). The temperature is unknown. Let's express it in the form of a formula. How to do it, I hope you know? Such assignments for the GIA in grade 9 usually give:

Now it remains to substitute numbers instead of letters on the right side and simplify:

Here's the answer: degrees Kelvin! And what a terrible task it was!

We continue to torment problems in physics.

Task 4

The height above the ground of a ball tossed up changes according to the law, where is the height in meters, is the time in seconds that has elapsed since the throw. How many seconds will the ball be at a height of at least three meters?

Those were all the equations, but here it is necessary to determine how much the ball was at a height of at least three meters, which means at a height. What are we going to make? Inequality, yes! We have a function that describes how the ball flies, where is exactly the same height in meters, we need the height. Means

And now you just solve the inequality, most importantly, do not forget to change the inequality sign from more or equal to less or equal when you multiply by both parts of the inequality in order to get rid of the minus in front.

Here are the roots, we build intervals for inequality:

We are interested in the interval where the sign is minus, since the inequality takes negative values ​​there, this is from to both inclusive. And now we turn on the brain and think carefully: for inequality, we used an equation that describes the flight of the ball, it somehow flies along a parabola, i.e. it takes off, reaches a peak and falls, how to understand how long it will be at a height of at least meters? We found 2 turning points, i.e. the moment when it soars above meters and the moment when it reaches the same mark while falling, these two points are expressed in our form in the form of time, i.e. we know at what second of the flight it entered the zone of interest to us (above meters) and into which it left it (fell below the meter mark). How many seconds was he in this zone? It is logical that we take the time of exit from the zone and subtract from it the time of entry into this zone. Accordingly: - so much he was in the zone above meters, this is the answer.

You are so lucky that most of the examples on this topic can be taken from the category of problems in physics, so catch one more, it is the final one, so push yourself, there is very little left!

Task 5

For a heating element of a certain device, the temperature dependence on the operating time was experimentally obtained:

Where is the time in minutes. It is known that at a temperature of the heating element above the device may deteriorate, so it must be turned off. Find the maximum time after the start of work to turn off the device. Express your answer in minutes.

We act according to a well-established scheme, everything that is given, we first write out:

Now we take the formula and equate it to the temperature value to which the device can be heated as much as possible until it burns out, that is:

Now we substitute numbers instead of letters where they are known:

As you can see, the temperature during operation of the device is described by a quadratic equation, which means that it is distributed along a parabola, i.e. the device heats up to a certain temperature, and then cools down. We received answers and, therefore, during and during minutes of heating, the temperature is critical, but between and minutes it is even higher than the limit!

So, you need to turn off the device after a minute.

MATHEMATICAL MODELS. BRIEFLY ABOUT THE MAIN

Most often, mathematical models are used in physics: after all, you probably had to memorize dozens of physical formulas. And the formula is the mathematical representation of the situation.

In the OGE and the Unified State Examination there are tasks just on this topic. In the USE (profile) this is task number 11 (formerly B12). In the OGE - task number 20.

The solution scheme is obvious:

1) From the text of the condition, it is necessary to “isolate” useful information - what we write in physics problems under the word “Given”. This useful information is:

• Formula
• Known physical quantities.

That is, each letter from the formula must be assigned a certain number.

2) Take all the known quantities and substitute them into the formula. The unknown value remains as a letter. Now you just need to solve the equation (usually quite simple), and the answer is ready.

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

It's like in sports - you need to repeat many times to win for sure.

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