Relationships between quantities characterizing the radiation field (energy flux density φ or particles φ N) and quantities characterizing the interaction of radiation with the environment (dose, dose rate) can be established by introducing the concept of mass energy transfer coefficient μ nm. It can be defined as the fraction of radiation energy transferred to a substance when passing through protection of a unit mass thickness (1 g/cm2 or 1 kg/m2). In the event that radiation with an energy flux density φ falls on the protection, the product φ · μ nm will give the energy transferred to a unit mass of a substance per unit time, which is nothing more than the absorbed dose rate:
P = φ μ nm (23)
P = φ γ E γ μ nm (24)
To go to the exposure dose rate, which is equal to the charge formed by gamma radiation per unit mass of air per unit time, it is necessary to divide the energy calculated using formula (24) by the average energy of formation of one pair of ions in the air. and multiply by the charge of one ion, equal to charge electron qe. In this case, it is necessary to use the mass energy transfer coefficient for air.
P 0 = φ γ E γ μ nm (25)
Knowing the relationship between gamma radiation flux density and exposure dose rate, it is possible to calculate the latter from a point source of known activity.
Knowing the activity A and the number of photons per 1 decay event n i, we obtain that per unit time the source emits n i · A photons in an angle of 4π.
To obtain the flux density at a distance R from the source, it is necessary to divide the total number of particles by the area of a sphere of radius R:
Substituting the resulting value of φ γ into formula (25) we obtain
Let us reduce the values determined from reference data for a given radionuclide into one coefficient K γ – gamma constant:
As a result, we obtain the calculation formula
When calculated in non-system units, the quantities have the following dimensions: R O – R/h; A – mCi; R – cm; Kγ – (R cm 2)/(mCi h);
in the SI system: P O – A/kg; A – Bk; R – m; Kγ – (A m2)/(kg Bq).
Relationship between gamma constant units
1 (A m 2)/(kg Bq) = 5.157 10 18 (R cm 2)/(h mCi)
Formula (29) has a very great importance in dosimetry (such as the Ohm's law formula in electrical and electronics engineering) and therefore must be memorized. The Kγ values for each radionuclide are found in the reference book. As an example, we present their values for nuclides used as control sources of dosimetric instruments:
for 60 Co Kγ = 13 (R cm 2)/(h mCi);
for 137 C Kγ = 3.1 (P cm 2)/(h mCi).
The given relationships between units of activity and dose rate made it possible to introduce such units of activity for gamma emitters as kerma equivalent and radium gamma equivalent.
Kerma equivalent is the amount of radioactive substance that, at a distance of 1 m, creates a kerma power in the air of 1 nGy/s. The unit of measurement for kerma equivalent is 1 nGym 2 /s.
Using the relationship according to which 1Gy=88R in air, we can write 1nGym2/s=0.316 mRm2/hour
Thus, the kerma equivalent of 1 nGym 2 /s creates an exposure dose rate of 0.316 mR/hour at a distance of 1 m.
The unit of radium gamma equivalent is the amount of activity that produces the same gamma dose rate as 1 mg of radium. Since the gamma constant of radium is 8.4 (Рּcm 2)/(hourּmKu), then 1 mEq of radium creates a dose rate of 8.4 R/hour at a distance of 1 m.
The transition from the activity of substance A in mKu to the activity in mEq of radium M is carried out according to the formula:
Ratio of kerma equivalent units to radium gamma equivalent units
1 mEq Ra = 2.66ּ10 4 nGym 2 /s
It should also be noted that the transition from exposure dose to equivalent dose and then to the effective dose of gamma radiation during external irradiation is quite difficult, because This transition is influenced by the fact that vital organs are shielded by other parts of the body during external irradiation. This degree of shielding depends both on the energy of the radiation and its geometry - from which side the body is irradiated - front, back, side or isotropically. Currently, NRBU-97 recommends using the transition 1Р=0.64 cSv, however, this leads to an underestimation of the doses taken into account and, obviously, appropriate instructions for such transitions have to be developed.
At the end of the lecture, it is necessary to return once again to the question - why five different quantities and, accordingly, ten units of measurement are used to measure doses of ionizing radiation. Accordingly, six units of measurement are added to them.
The reason for this situation is that different physical quantities describe different manifestations of ionizing radiation and serve different purposes.
The general criterion for assessing the danger of radiation to humans is the effective equivalent dose and its dose rate. It is this that is used to standardize exposure under the Radiation Safety Standards of Ukraine (NRBU-97). According to these standards, the dose limit for personnel nuclear power plants and institutions working with sources of ionizing radiation is 20 mSv/year. For the entire population – 1 mSv/year. Dose equivalent is used to assess the effects of radiation on individual organs. Both of these concepts are used in normal radiation conditions and in minor accidents when doses do not exceed five permissible annual dose limits. In addition, the absorbed dose is used to assess the effect of radiation on a substance, and the exposure dose is used to objectively assess the gamma radiation field.
Thus, in the absence of large nuclear accidents To assess the radiation situation, we can recommend a dose unit - mSv, a dose rate unit μSv/hour, an activity unit - Becquerel (or off-system rem, rem/hour and mKu).
The appendices to this lecture provide relationships that may be useful for orientation in this problem.
Appendix No. 1
✓ What is called a physical quantity?
✓ Give examples of the relationship between physical quantities.
1. As you know, physical quantities are used to describe physical phenomena and properties of bodies and substances.
While conducting experiments, scientists noticed that the quantities that characterize the same phenomenon are mutually related.
For example, when the temperature of bodies changes, their volume and length change. They increase as the temperature rises and decrease as the temperature decreases. The temperature of the water in the kettle when heated depends on the heating time.
2. To conclude that the relationship between quantities is not accidental, its validity is checked for many similar phenomena.
If connections between quantities characterizing a phenomenon appear constantly, then they are called physical laws.
There are physical laws relating only to certain physical phenomena. For example, there are laws that describe mechanical phenomena, or laws that govern thermal phenomena. In addition, there are more general laws that are valid for all physical phenomena. The set of phenomena that are described by laws is determined by the limits of their applicability.
Of course, a physical law is written in the form of a formula.
3. Knowledge of the surrounding world would be incomplete if people only observed and described phenomena and established laws. You also need to be able to explain natural phenomena. When studying nature, a person always strives to answer not only the question “What is happening?” but also to the question “Why is this happening?”
The answer to the question “Why does this or that phenomenon occur?” can be obtained with the help of theoretical knowledge, which is the basis of physical theory. Thus, mechanical phenomena, for example, the nature of the movement of vehicles or Earth satellites, are explained by a theory called mechanics. The molecular kinetic theory of the structure of matter makes it possible to explain why bodies expand when heated, why a spoon placed in a glass of hot tea heats up. There are theories that explain electrical, optical and magnetic phenomena.
Thus, physical phenomena - mechanical, thermal, electrical and others - are explained by the corresponding physical theories. Theory contains general, systematized knowledge about physical phenomena.
The theory allows not only to explain why a phenomenon occurs, but also to predict its course.
Self-test questions
1. What does a physical law express?
3. What is the role of physical theory?
4. What phenomena does mechanics explain?
Lesson on the topic "Connections between quantities. Function»
Yumaguzhina Elvira Mirkhatovna,
teaching experience 14 years,
1st qualification category, MBOU "Barsovskaya Secondary School No. 1",
UMK:"Algebra. 7th grade",
A.G.Merzlyak, V.B.Polonsky, M.S.Yakir,
"Ventana-Graf", 2017.
Didactic rationale.
Lesson type: Lesson on learning new knowledge.
Teaching aids: PC, multi-projector.
Educational: learn to determine the functional relationship between quantities, introduce the concept of function.Developmental: develop mathematical speech, attention, memory, logical thinking.
Planned result
Subject
skills
UUD
form the concepts of functional dependence, function, function argument, function value, domain of definition and domain of function.
Personal: develop the ability to plan your actions in accordance with the educational task.
Regulatory: develop students’ ability to analyze, draw conclusions, determine relationships and logical sequence of thoughts;
train the ability to reflect on one’s own activities and the activities of one’s friends.
Cognitive: analyze, classify and summarize facts, build logical reasoning, use demonstrative mathematical speech.
Communicative: independently organize interaction in pairs, defend your point of view, give arguments, confirming them with facts.
Basic Concepts
Dependency, function, argument, function value, scope and scope.
Organization of space
Interdisciplinary connections
Forms of work
Resources
Algebra - Russian language
Algebra - physics
Algebra - Geography
Frontal
Individual
Work in pairs and groups
Projector
Textbook
Self-assessment sheet
Lesson stage
Teacher activities
Planned student activities
Developed (formed) learning activities
subject
universal
1.Organizational.
Slide 1.
Slide 2.
Greeting students; teacher checking the class's readiness for the lesson; organization of attention.
What do a climber storming the mountains have in common with a child successfully playing computer games, and a student striving to learn better and better.
Get ready for work.
Result of success
Personal UUD: the ability to highlight the moral aspect of behavior
Regulatory UUD: the ability to reflect on one’s own activities and the activities of comrades.
Communicative UUD
Cognitive UUD: conscious and voluntary construction of a speech utterance.
2. Setting the goals and objectives of the lesson. Motivation for students' learning activities.
Slide 2.
Everything in our lives is interconnected, everything that surrounds us depends on something. For example,
What does your current mood depend on?
What do your grades depend on?
What determines your weight?
Determine what is the keyword of our topic? Is there a relationship between objects? We will introduce this concept in today's lesson.
Interact with the teacher during oral questioning.
Addiction.
Write down the topic “Relationship between quantities”
Personal UUD:
development of motives for educational activities.
Regulatory UUD: decision-making.
Communicative UUD: listen to the interlocutor, construct statements that are understandable to the interlocutor.
Cognitive UUD: building a strategy for finding solutions to problems. Highlight essential information, put forward hypotheses and update personal life experience
3. Updating knowledge.
Work in pairs.
Slide 3.
Slide 4.
You have tasks on your tables that need to be solved in pairs.
Calculate the value of y using the formula y = 2x+3 for a given value of x.
Annex 1.
Writes down students' answers at their desks under dictation for verification, matching the meanings of expressions and letters from students' cards in ascending order.
Appendix 2.
Shows a collage of famous mathematicians who first worked on the “function”.
Give your calculations.
They voice their answers, check the solution, write out the correspondence of the letters from the cards with the obtained values in ascending order.
- "Function"
Perception of information.
Repeating value calculations literal expressions with a known value of one variable, working with integers in ascending order. Identification of a new concept of “function”.
Personal UUD:
Adoption social role student, meaning formation.
Regulatory UUD: drawing up a plan and sequence of actions, predicting the result and level of mastery of the material,searching and retrieving the necessary information,building a logical chain of reasoning, proof.
Cognitive UUD: the ability to consciously construct a speech utterance.
Communicative skills: the ability to listen to the interlocutor,conducting dialogue, observing moral standards when communicating.
4. Primary assimilation of new knowledge.
Group.
Slide 5.
Organizes the perception of information by students, comprehension of the given and primary memorization by children of the topic being studied: “Relationship between quantities. Function". Organizes work in groups (4 people) on cases.
Each group has a case with assignments on the table. Conditions modern life dictate their own rules and one of these rules is to have your own cellular telephone. Let's consider a real-life example when we use cellular communications at the MTS tariff "Smartmini».
Appendix 3.
Guides groups in decision making.
Distribute tasks in the group.
Ability to listen to a task, understand how to work with a case: analysis of the dependence of one variable on another, introduction of new definitions “Function, argument, domain of definition”, work with the graph “Dependence of telephone charges”
Personal UUD:
Regulatory UUD: monitoring the correctness of answers to information from the textbook, developing students’ own attitude to the material studied, correcting perception.
Cognitive UUD: search and selection necessary information.
Communication UUD:
listen to the interlocutor, construct statements that are understandable to the interlocutor. Meaningful reading.
5. Initial check of understanding. Individual.
Slide 6.
Organizes student responses.
Case protection
The ability to prove the correctness of your decision.
Personal UUD: development of cooperation skills.
Regulatory UUD: developing students’ own attitude towards the studied material,use demonstrative mathematical language.
Communicative UUD: the ability to listen and intervene in front of students, listen to the interlocutor, and construct statements that are understandable to the interlocutor.Cognitive UUD: search and selection of necessary information, the ability to read function graphs, justify one’s opinion;
6. Primary consolidation. Frontal.
Slide 7.
Organizes work according to a common task.
Determines the relationship between algebra and physics, algebra and geography.
Appendix 4.
Answer the teacher's questions and read the schedule.
Ability to apply previously learned material.
Personal UUD:
independence and critical thinking.
Regulatory UUD: carry out self-monitoring of the task completion process. Correction.
Cognitive UUD: compare and summarize facts, build logical reasoning, use demonstrative mathematical speech.
Communication UUD:
meaningful reading.
7. Information about homework, instructions on how to complete it.
Slide 8.
Explains homework.
Level 1 – mandatory. §20, questions 1-8, No. 157, 158, 159.
Level 2 – intermediate. Select examples of the dependence of one quantity on another from any branch of life.
Level 3 – advanced. Analyze the functional dependence of payment utilities, derive a formula for calculating any service, build a graph of the function.
Plan their actions in accordance with self-esteem.
Working at home with text.
Know the definitions on the topic, formulate a relationship through a formula, and the ability to build a relationship between one quantity and another.
Personal UUD:
acceptance of the social role of the student.
Regulatory UUD:adequately carry out self-assessment, correction of knowledge and skills.
Cognitive UUD:carry out updating of acquired knowledge in accordance with the level of assimilation.
8. Reflection.
Slide 9.
Organizes a discussion of achievements and instructions on how to use the self-assessment sheet. Offers self-assessment of achievements by filling out a self-assessment sheet.
Appendix 5.
Familiarization with the self-assessment sheet, clarification of evaluation criteria. They draw conclusions and self-assess their achievements.
Conversation to discuss achievements.
Personal UUD:
independence and critical thinking.
Regulatory UUD: accept and save the educational goal and task, carry out final and step-by-step control based on the result, plan future activities
Cognitive UUD: analyze the degree of assimilation of new materialCommunicative UUD: listen to classmates, voice their opinions.
Annex 1.
Answers for the teacherfor check
Match the answers for a new concept in ascending order of meaning
Calculate the value of y using the formula y=2x+3 if x = 2
Calculate the value of y using the formula y=2x+3 if x = -6
Calculate the value of y using the formula y=2x+3 if x = 4
Calculate the value of y using the formula y=2x+3 if x = 5
Calculate the value of y using the formula y=2x+3 if x = -3
Calculate the value of y using the formula y=2x+3 if x = 6
Calculate the value of y using the formula y=2x+3 if x = -1
Calculate the value of y using the formula y=2x+3 if x = -5
Calculate the value of y using the formula y=2x+3 if x = 0
Calculate the value of y using the formula y=2x+3 if x = - 2
Calculate the value of y using the formula y=2x+3 if x = 3
Calculate the value of y using the formula y=2x+3 if x = -4
Appendix 2.
Appendix 3.
(2 people)In tariff cellular communication «
Smartmini» includes not only a subscription fee of 120 rubles, but also a fee for a conversation per minute with other Russian cellular operators, each minute of conversation is equal to 2 rubles.
1. Let’s calculate the telephone fee for a month if we had a conversation through another mobile operator for 2 minutes, 4 minutes, 6 minutes, 10 minutes
Write down an expression to calculate the telephone fee for 2min, 4min, 6min, 10min.
Withdraw general formula to calculate telephone charges.
S = 120 + 2∙2 = 124rub.
S = 120 + 2∙4 = 128rub.
S = 120 + 2∙6 =132rub.
S = 120 + 2∙8 = 136rub.
S = 120 + 2∙10 = 140rub.
S = 120 + 2∙t
Task No. 2
(2 people)
Working with the textbook. Define the following concepts
Function –
Function argument -
Domain -
Range of values -
This is a rule that allows you to find a single value for the dependent variable for each value of the independent variable.
Independent variable.
These are all the values that the argument takes.
This is the value of the dependent function.
Task No. 3
(4 people). In the “Telephone fee dependence” card, mark the fee values at 4 minutes, 6 minutes, 8 minutes, 10 minutes with a dot. (Take the values from task No. 1).
Attention! Telephone fee value at 2 min. already installed.
"Phone Charge Dependency"
Determine the domain of definition and the domain of value of the function from the graph
Range of definition – from 2 to 10
Range of values – from 124 to 140
Appendix 4.
Appendix 5.
Self-assessment sheet
Self-esteemCriteria for assessing a classmate at a desk
Classmate's rating (F.I.)
Formulation of the lesson topic, purpose and objectives of the lesson.
I was able to determine the topic, purpose and objectives of the lesson - 2 points.
I was able to determine only the topic of the lesson - 1 point.
I could not determine the topic, purpose and objectives of the lesson - 0 points.
Participated in determining the topic of the lesson, the purpose of the lesson, or the objectives of the lesson - 1 point.
Did not participate in determining the topic of the lesson, the purpose of the lesson, or the objectives of the lesson 0 b
What will I do to achieve the goal.
I myself determined how to achieve the goal of the lesson - 1 point.
I could not determine how to achieve the lesson goal - 0 points.
Participated in planning actions to achieve the lesson goal - 1 point.
Did not participate in planning actions to achieve the lesson goal 0 b
Performance practical work paired with.
Participated in group work – 1 point.
Did not participate in the work of the group - 0 point.
Working in a group to work on a case.
Participated in group work – 1 point.
Did not participate in the work of the group - 0 point.
Participated in group work – 1 point.
Did not participate in the work of the group - 0 point.
Performing a task with function graphs.
I made all the examples myself -2 points.
Did less than half myself - 0 points.
Completed the task at the board 1 point.
Didn't complete the task on the board 0 points.
Choosing homework
3 points - chose 3 tasks out of 3, 2 points - chose only 2 numbers, 1 point - chose 1 task out of 3
Not Evaluated
Give yourself a rating: if you scored 8-10 points - “5”; 5 – 7 points – “4”; 4 – 5 points – “3”.
Self-analysis of the lesson.
This lesson is No. 1 in the system of lessons on the topic “Function”.
The purpose of the lesson is to form an idea of the function, how mathematical model descriptions of real processes. The main activities of the student are repetition of computational skills with whole expressions, formation of primary ideas about the relationships between quantities, description of the concepts of “function, dependent variable”, “argument, independent variable”, distinguishing functional dependencies among dependencies in the form of a function graph.
Developmental: develop mathematical speech (use of special mathematical terms), attention, memory, logical thinking, draw conclusions.
Educational: to cultivate a culture of behavior during frontal, group, pair and individual work, to form positive motivation, to cultivate the ability to self-esteem.
The type of this lesson is a lesson in mastering new knowledge; it includes seven stages. The first stage is organizational, the mood for educational activities. The second stage is the motivation of educational activities to set goals and objectives for the lesson “Relationships between quantities. Function". The third stage is updating knowledge, working in pairs. The fourth stage is the initial assimilation of new knowledge, “case technology”, work in a group. Fifth stage - initial check of understanding - individual work, case protection. The sixth stage - primary consolidation - frontal work, discord of examples of function graphs. The seventh stage – information about homework, instructions on how to complete it in an individual form of 3 levels. The eighth stage is reflection, summing up, filling out a self-assessment sheet by students about personal achievements in the lesson.
When motivating students for the lesson, I selected cases from life, where connections between quantities were considered not only in life, but also connections in algebra, physics, and geography. Those. The assignments were focused on creative thinking, resourcefulness, and on strengthening the applied orientation of the algebra course by considering examples of real relationships between quantities based on the students’ experience, which helped to ensure that all students understood the material.
I managed to meet the deadline. Time was distributed rationally, the pace of the lesson was high. The lesson was easy to teach; the students quickly got involved in the work and gave interesting examples of relationships between quantities. During the lesson, an interactive whiteboard was used, accompanied by a presentation of the lesson. I think the goal of the lesson has been achieved. As reflection showed, the students understood the lesson material. Homework did not cause any difficulty. Overall, I think the lesson was successful.
Between physical quantities There are qualitative and quantitative dependencies, a natural connection, which can be expressed in the form of mathematical formulas. The creation of formulas is associated with mathematical operations on physical quantities.
Homogeneous quantities admit all types of algebraic operations on themselves. For example, you can add the lengths of two bodies; subtract the length of one body from the length of the second; divide the length of one body by the length of the second; raise length to power. The result of each of these actions has a specific physical meaning. For example, the difference in the lengths of two bodies shows how much longer the length of one body is than the other; the product of the base of the rectangle and the height determines the area of the rectangle; the third power of the length of an edge of a cube is its volume, etc.
But it is not always possible to add two quantities of the same name; for example, the sum of the densities of two bodies or the sum of the temperatures of two bodies are devoid of physical meaning.
Dissimilar quantities can be multiplied and divided by each other. The results of these actions on heterogeneous quantities also have a physical meaning. For example, the product of the mass m of a body and its acceleration a expresses the force F under the influence of which this acceleration was obtained, that is:
the quotient of dividing the force F by the area S on which the force acts uniformly expresses the pressure p, that is:
In general, the physical quantity X can be expressed using mathematical operations in terms of other physical quantities A, B, C, ... by an equation of the form:
(1.6)
where is the proportionality coefficient.
Exponents 
can be either integer or fractional, and can also take a value equal to zero.
Formulas of the form (1.6), which express one physical quantity in terms of another, are called equations between physical quantities.
The coefficient of proportionality in equations between physical quantities, with rare exceptions, is equal to unity. For example, an equation in which the coefficient differs from unity is the equation of the kinetic energy of a body in translational motion:
. (1.7)
The value of the proportionality coefficient, both in this formula and in general in equations between physical quantities, does not depend on the choice of units of measurement, but is determined solely by the nature of the relationship between the quantities included in this equation.
The independence of the proportionality coefficient from the choice of units of measurement is a characteristic feature of equations between quantities. That is, each of the symbols A, B, C, ... in this equation represents one of the specific implementations of the corresponding quantity, which does not depend on the choice of unit of measurement.
But if all the quantities included in equation (1.6) are divided into the appropriate units of measurement, we obtain an equation of a new type. For simplicity of consideration, we write the following equation:
After dividing the quantities X, A and B by their units of measurement, we obtain:
, (1.9)
. (1.10)
Equations of the form (1.9) or (1.10) no longer connect quantities as collective concepts, but their numerical values obtained as a result of expressing quantities in certain units of measurement.
An equation that relates numerical values of quantities is called an equation between numerical values.
For example, the numerical value of heat Q, which is released in a conductor during the passage of current:
, (1.11)
where is the numerical value of the heat that is released on the conductor, kcal; numerical value of current, A; numerical value of resistance, Ohm; numerical value of time, s.
Only under these conditions does the numerical coefficient take on the value of 0.24.
But in technical calculations, such equations are used very widely. Values are expressed in different systems and non-systemic units, thereby obtaining equations with complex coefficients.
In general, the proportionality coefficient in equations between numerical values depends only on the units of measurement. Replacing the unit of measurement of one or more quantities included in equation (1.9) entails a change in the numerical value of the coefficient.
The dependence of the proportionality coefficient on the choice of units of measurement is distinctive feature equations between numerical values. This characteristic feature between numerical values is used to determine derived units of measurement and to construct systems of units.
Let's transfer the gable data. 6.2 on the graph (Fig. 6.1). By plotting labor costs on the horizontal axis and output volume on the vertical axis, we can construct curves for total, average and marginal products. Graphically the value MR is determined by the tangent of the angle of inclination of the tangent to the curve of the total product at the point corresponding to a certain volume of it, the value AR- tangent of the angle of inclination of the ray going from the origin to the same point.
When constructing the marginal product curve, the corresponding values MR must be put aside in the middle of segment D L(If MR= 39, then on the graph this value is plotted at L = 2,5).
As follows from the table. 6.2 and graphs in Fig. 6.1, I and B, the introduction of additional units of a variable resource (in our case, labor) at a fixed value of capital leads to a constant increase in the total product TR. However, a more thorough analysis shows that this growth occurs unevenly: in the section (O - L t), the increments of DTR at the same increments of DL increase (curve TR has a concave appearance), and with a further increase in the number of units of labor used, the increment Asia-Pacific contract (the GR curve becomes convex).
Rice. 6.1. Total curves (A), average and extreme (b)
product
Such a change in the volume of output of goods and services depending on the increase in input units of a variable factor reflects the action of one of the fundamental laws of economics - the law of diminishing returns of a resource. According to this law the introduction of additional units of a variable resource with a constant value of a constant factor will certainly lead to a situation where each subsequent unit of a variable factor begins to add less to the total product than its previous unit.
This is tantamount to saying that under the conditions listed above, there will certainly come a time when a further increase in the units of the variable factor used will cause a decrease in the marginal product, therefore this law is sometimes called the law of an inevitable decrease in the marginal product.
The general meaning of the law of diminishing returns is that the use of a constant factor in the production of a good limits the increase in the volume of output of this good with a consistent increase in the number of units of the variable resource used.
How can we explain the law of diminishing returns for a resource? With one fixed factor (capital), the introduction of additional units of the variable factor (labor) at first (section OL() makes it possible to effectively use the division of labor. This results in each additional worker producing everything large quantity goods and services, i.e. marginal product increases. However, at some point, the next worker will become redundant - all possibilities for the division of labor have been exhausted, and he will have to wait until the machine is free to apply his labor. From this point on, the services of each subsequent worker will be increasingly useless, causing a further decline in marginal product. Theoretically, a situation may arise where an additional worker begins to interfere with production, and this will cause a decrease in production volumes. In this case, the values of the marginal product will become negative and the curve MR will cross the x-axis, and the curve TR will decrease (hypothetically similar situation happens at the point L 3 in Fig. 6.1, A And b).
Undoubtedly, this law can also be interpreted as the law of an inevitable decline in the average product, since under similar conditions there will certainly come a moment when further growth units of the variable factor used will lead to a decrease in the average product.
Example 2. Suppose that 2 workers take part in the production of 42 units of goods, who produce on average 21 units of goods per month, i.e. LR = TP/L= 42/2 = 21. Let the company hire one more, third worker. If the return of an additional worker hired (i.e., the marginal product) is higher than the average yield of each of the available workers, for example, 39 units, then the value of the average product, taking into account the hiring of three workers, will be more than 21 units:
This means that as long as MP > AR, i.e. the value of the marginal product exceeds the value of the average product, the latter increases; in this case, on the graph (see Fig. 6.1) the marginal product curve is located above the average product curve. If MR and the marginal product curve passes below the average product curve, then the value AR decreases. Therefore, the curve MR crosses the curve AR at the point where the curve AR has a maximum.
