Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify a specific person or contact him.
You may be asked to provide your personal information at any time when you contact us.
The following are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you submit an application on the site, we may collect various information, including your name, phone number, address Email etc.
How we use your personal information:
- Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send you important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.
Disclosure to third parties
We do not disclose information received from you to third parties.
Exceptions:
- In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest purposes.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.
Maintaining your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.
This is the simplest and most accurate homogeneous difference scheme for calculating gas dynamics. Its template is shown in Fig. 98; the values of the radii are assigned to the nodes of the grid, the values of the velocity are assigned to the boundaries of spatial intervals on half-integer layers, and the values of density, pressure, and internal energy are assigned to the midpoints of the intervals on entire layers.
The construction of the circuit resembles an acoustic "cross". For ease of notation, we choose steps and t uniform in mass and time and approximate the system by the following difference equations:
These equations are written in the order that is convenient for calculations.
Let us discuss the difference expression for viscous pressure (65). To carry out the limiting transition from the difference scheme to the equations of gas dynamics, one must first tend to zero at a fixed viscosity coefficient, and then construct a series of such limiting solutions for infinitely decreasing values of . But it's very labor intensive. Therefore, in practice, these limit transitions are combined into one common one, assuming that although the legality of such a procedure has not been proven (the density is introduced into the formula so that the coefficients are dimensionless).
Thus, viscous pressure (65) takes the form
where is the speed of sound. Expression (67) is written for the plane case; but usually it is used for any symmetry of the problem.
Approximation. From the template view in Fig. 98 and the symmetrical writing of scheme (66), it is easy to see that on flows without compression, when pseudoviscosity (67) vanishes, the “cross” scheme has a local approximation
On flows with compression (including shock waves), the pseudoviscosity is nonzero. True, the quadratic term in (67a) has a value, but the linear term has a value and, thus, worsens the order of approximation. In addition, the viscous terms are not written quite symmetrically in time. As a result, the approximation deteriorates to
Finding a difference solution. Scheme (66) - explicit; calculations are carried out in the following way. Let all quantities on the initial layer be known. Then from the momentum difference equation (66a) we find in all intervals; then from the second equation (66b) we determine and from equation (66c) - .
The energy equation (66d) is solved last. Formally, it is an implicit algebraic equation for definition in a given interval. But for each value of the index, equations (66d) are solved independently without forming a coupled system of equations, so that the difference scheme essentially remains explicit.
Remark 1. The energy equation in (66) can be made explicit using only the value from the original layer:
This somewhat simplifies the calculation, does not affect the stability, but noticeably worsens the accuracy, since the approximation error becomes even for smooth flows. This option is rarely used.
The stability of the circuit can be investigated by the method of separation of variables, by linearizing the circuit and freezing the coefficients. Cumbersome calculations lead to a Courant-type stability condition.
For example, on smooth flows with zero viscosity, the scheme is stable at
For an ideal gas, and condition (69) takes the form where is the adiabatic speed of sound. On flows with non-zero viscosity, the limitation on the step is somewhat stronger; at quadratic viscosity, the stability condition takes the form
where is the velocity jump on the shock wave. Although this study is not rigorous, nevertheless, this stability condition is well confirmed in practice.
Thus, the "cross" is a conditionally stable scheme. We note an interesting circumstance. Viscosity is not needed to calculate smooth flows. And if we calculate the shock wave without viscosity (choosing a small one that satisfies condition (70)), then we get the “blank pattern” shown in Fig. 99. This calculation is stable because the amplitude of the oscillations does not increase with time. But there is no convergence to a physically correct solution for , since the approximation is lost on the discontinuity.
The convergence of the gas-dynamic scheme "cross" has not been proven. However, this scheme has been successfully used in calculations since about 1950 and has been tested on many difficult problems with known exact solutions. As the steps tend to zero, convergence to the correct solution was observed if the steps satisfied the stability condition.
Remark 2. Scheme (66) is non-conservative; however, its imbalance tends to zero at
Remark 3. Gas-dynamic problems with very thin layers are especially difficult to calculate. Indeed, if , then in order to calculate with satisfactory accuracy according to formula (66c), it is necessary to know the radii with a very high accuracy, comparable to rounding errors on a computer. In such problems, sometimes it is necessary to carry out calculations with a double number of digits or to modify the difference scheme on purpose.
The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Equations have been used by man since ancient times and since then their use has only increased. If you see an expression with fractions with a variable in the numerator / denominator, then you have an expression called in mathematics a rational equation. In general, we can call rational equations all equations that have 1 rational expression in their composition. As for the solutions of rational equations, they are solved as follows: operations are performed on the left and right sides until the moment when the variable is isolated on one side. There are two ways to solve such equations:
Multiplication crosswise;
LCV (lowest common denominator).
The first method is used if, after the equation has been rewritten, one fraction has formed on each side of it. For example:
\[\frac (x+3)(4)- \frac(x)(2)= 0\]
To use the crosswise multiplication method, you need to convert the equations to the form:
\[\frac (x+3)(4)= \frac (x)(-2)\]
The second method can be used when you have an equation with 3/more fractions. For example:
\[\frac (x)(3)+ \frac (1)(2)=\frac(3x+1)(6) \]
For this equation, the least common multiple will be 6, which will make it easy to solve this equation.
Where can I solve a rational equation online for free?
You can solve a rational equation online with a solution on our website https: // site. Free online solver will allow you to solve an online equation of any complexity in seconds. All you have to do is just enter your data into the solver. You can also watch the video instruction and learn how to solve the equation on our website. And if you have any questions, you can ask them in our Vkontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.
Today we continue a series of video tutorials on percentage problems from the Unified State Examination in mathematics. In particular, we will analyze two very real problems from the Unified State Examination and once again see how important it is to carefully read the condition of the problem and interpret it correctly.
So the first task is:
A task. Only 95% and 37,500 graduates of the city solved problem B1 correctly. How many people correctly solved problem B1?
At first glance, it seems that this is some kind of task for the caps. Like:
A task. There were 7 birds on the tree. 3 of them flew away. How many birds have flown?
However, let's do the math. We will solve by the method of proportions. So, we have 37,500 students - this is 100%. And also there is a certain number x of students, which is 95% of the very lucky ones who correctly solved problem B1. We write it down:
37 500 — 100%
X - 95%
You need to make a proportion and find x. We get:
Before us is a classic proportion, but before using the main property and multiplying it crosswise, I propose to divide both parts of the equation by 100. In other words, we cross out two zeros in the numerator of each fraction. Let's rewrite the resulting equation:
According to the basic property of proportion, the product of the extreme terms is equal to the product of the middle terms. In other words:
x = 375 95
These are quite large numbers, so you have to multiply them by a column. I remind you that it is strictly forbidden to use a calculator on the exam in mathematics. We get:
x = 35625
Total answer: 35,625. That is how many people out of the original 37,500 solved problem B1 correctly. As you can see, these numbers are pretty close, which makes sense because 95% is also very close to 100%. In general, the first task is solved. Let's move on to the second.
Interest problem #2
A task. Only 80% of the city's 45,000 graduates solved problem B9 correctly. How many people solved problem B9 incorrectly?
We solve in the same way. Initially, there were 45,000 graduates - this is 100%. Then, x graduates must be selected from this number, which should be 80% of the original number. We make a proportion and solve:
45 000 — 100%
x - 80%
Let's reduce one zero in the numerator and denominator of the 2nd fraction. Let's rewrite the resulting construction once more:
The main property of proportion: the product of the extreme terms is equal to the product of the middle ones. We get:
45,000 8 = x 10
It's the simplest linear equation. Let's express the variable x from it:
x = 45,000 8:10
We reduce one zero at 45,000 and at 10, the denominator remains one, so all we need is to find the value of the expression:
x = 4500 8
You can, of course, do the same as last time, and multiply these numbers in a column. But let's not make life difficult for ourselves, and instead of multiplying by a column, we decompose the eight into factors:
x = 4500 2 2 2 = 9000 2 2 = 36,000
And now - the most important thing that I talked about at the very beginning of the lesson. You need to carefully read the condition of the problem!
What do we need to know? How many people solved problem B9 not right. And we just found those people who decided correctly. These turned out to be 80% of the original number, i.e. 36,000. This means that in order to get the final answer, our 80% must be subtracted from the original number of students. We get:
45 000 − 36 000 = 9000
The resulting number 9000 is the answer to the problem. In total, in this city, out of 45,000 graduates, 9,000 people solved problem B9 incorrectly. Everything, the task is solved.
I hope that this video will help those who are preparing for the exam in mathematics on their own. And that's all for me. Pavel Berdov was with you. See you soon! :)
Proportion Formula
Proportion is the equality of two ratios when a:b=c:d
ratio 1 : 10 is equal to the ratio of 7 : 70, which can also be written as a fraction: 1 10 = 7 70 reads: "one is to ten as seven is to seventy"Basic properties of proportion
The product of the extreme terms is equal to the product of the middle terms (crosswise): if a:b=c:d , then a⋅d=b⋅c
1 10 ✕ 7 70 1 ⋅ 70 = 10 ⋅ 7Proportion inversion: if a:b=c:d , then b:a=d:c
1 10 7 70 10 1 = 70 7Permutation of middle members: if a:b=c:d , then a:c=b:d
1 10 7 70 1 7 = 10 70Permutation of extreme members: if a:b=c:d , then d:b=c:a
1 10 7 70 70 10 = 7 1Solving a proportion with one unknown | The equation
1 : 10 = x : 70 or 1 10 = x 70To find x, you need to multiply two known numbers crosswise and divide by opposite meaning
x = 1 ⋅ 70 10 = 7How to calculate proportion
A task: you need to drink 1 tablet of activated charcoal per 10 kilograms of weight. How many tablets should be taken if a person weighs 70 kg?
Let's make a proportion: 1 tablet - 10 kg x tablets - 70 kg To find x, you need to multiply two known numbers crosswise and divide by the opposite value: 1 tablet x tablets✕ 10 kg 70 kg x = 1 ⋅ 70 : 10 = 7 Answer: 7 tablets
A task: Vasya writes two articles in five hours. How many articles will he write in 20 hours?
Let's make a proportion: 2 articles - 5 hours x articles - 20 hours x = 2 ⋅ 20 : 5 = 8 Answer: 8 articles
I can say to future school graduates that the ability to make proportions was useful to me both in order to proportionally reduce pictures, and in the HTML layout of a web page, and in everyday situations.
