Before we move on to the practical part of the article, where we will look for the volume of a parallelepiped, let's remember what kind of figure this is and find out why we may need these calculations.
There are three definitions, and they are all equivalent. So, a parallelepiped is:
1. A polyhedron with six faces, each of which is a parallelogram.
2. Hexagon, which has three pairs of faces parallel to each other.
3. A prism with a parallelogram at its base.
Perhaps the most common in our real life The types of geometric figure under consideration are a rectangular parallelepiped and a cube. In addition, a distinction is made between inclined and straight parallelepiped.
Rectangular parallelepiped: volume
A rectangular parallelepiped is distinguished by the fact that each face is a rectangle. As everyday example This figure can be used in an ordinary box (shoe box, gift box, postal box).
First, you need to find the values of the two sides of the base of the parallelepiped, which are located perpendicular to each other (on a plane they would be called width and length).
P = A*B, where A is length, B is width.
Now we make one more measurement - the height of the given figure, which we will call H.
Well, we find out the required volume if we multiply the height by the area of the base, that is:
Volume of a right parallelepiped
A straight parallelepiped is distinguished by the fact that its lateral faces are rectangles due to the fact that they are perpendicular to the bases of the figure.
The volume is calculated in a similar way, the only difference is that the height here is not an edge of the parallelepiped. IN in this case it represents a line that connects two opposite faces figure and perpendicular to its base.
Since the base of your parallelepiped is a parallelogram and not a rectangle, the formula for calculating the area of the base becomes somewhat more complicated. Now it will look like this:
P = A * B * sin(a), where A, B are the length and, accordingly, the width of the base, and “a” is the angle they form when they intersect.
How to find the volume of an inclined parallelepiped?
Any parallelepiped that is not straight is considered inclined.
Due to the fact that the edges of this figure are not perpendicular to the base, you first need to find the height. Multiplying it by the area of the base (see formula above), you get the volume:
V = P*H, where P is the base area, H is the height.
Volume of a parallelepiped with square edges
A cube is a rectangular parallelepiped, each of the six faces of which is a square. This implies the property of this figure - all its edges are equal to each other. As an example, let's imagine a children's toy like cubes.
Well, finding the volume of a cube is generally extremely simple. To do this, you only need to make one measurement (the edges) and raise the resulting value to the third power. Like this:
V = A³.
How can the volume of a parallelepiped be useful to us in life?
Let's say that you are puzzled by such a problem as the number of boxes that can fit in the trunk of your car. To do this, you need to arm yourself with a ruler or tape measure, a pen, a sheet of paper, as well as the above formulas rectangular parallelepiped.
By measuring the volume of one box and multiplying the value by the number of boxes you have, you will know how many cubic centimeters it will take to fit them in the trunk of your car.
And yes, remember that in some cases it will be advisable to convert cubic centimeters to meters. So, if as a result you received a box volume equal to 50 cm cubed, then to convert, simply multiply this figure by 0.001. This will give you cubic meters. And if you want to find out the volume in liters, then multiply the result in cubic meters by 1000.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. WITH physical point From a perspective, it looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Stay in constant units time measurements and do not go to reciprocals. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia logical paradox it can be overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Such absurd logic sentient beings never understand. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “screw me, I’m in the house”, or rather “mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Apply mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” We explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities mud, crystal structure and the arrangement of atoms in each coin is unique...
And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of numbers given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don't think this girl is stupid, no knowledgeable in physics. She just has an arch stereotype of perception graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
Volume of a parallelepiped
The size of the volume gives us an idea of what part of the space the object of interest to us occupies, and to find the volume of a rectangular parallelepiped we need to multiply its base area by its height.
IN Everyday life, most often to measure the volume of liquid, as a rule, they use the following measuring unit, as liter = 1dm3.
In addition to this unit of measurement, the following is used to determine volume:
A parallelepiped is one of the simplest three-dimensional figures and therefore finding its volume is not difficult.
Volume of a parallelepiped equal to the product its length, width and height. Those. To find the volume of a rectangular parallelepiped, it is enough to multiply all three of its dimensions.
To find the volume of a cube, you need to take its length and raise it to the third power.
Definition of a parallelepiped
Now let's remember what a parallelepiped is and how it differs from a cube.
A parallelepiped is called such three-dimensional figure, which has a polygon at its base. The surface of a rectangular parallelepiped consists of six rectangles, which are the faces of this parallelepiped. Therefore, it is logical that the parallelepiped has six faces, which consist of parallelograms. All faces of this polygon, which are located opposite each other, have the same dimensions.
All the edges of the parallelepiped are the sides of the faces. But the points of contact of the faces are the vertices of this figure.
Exercise:
1. Look carefully at the drawing and tell me what it reminds you of?
2. Think and answer where in everyday life you might encounter such a figure?
3. How many edges does the parallelepiped have?
Types of parallelepipeds
Parallelepipeds are divided into several varieties, such as:
Rectangular;
Inclined;
Cube
Rectangular parallelepipeds include those figures whose faces consist of rectangles.
If side faces are not perpendicular to its base, then in front of you is an inclined parallelepiped.
A figure such as a cube is also a parallelepiped. All of its faces, without exception, have the shape of squares.
Properties of a parallelepiped
The figure under study has a number of properties, which we will now learn about:
Firstly, opposite faces of this figure are equal and parallel to each other;
Secondly, it is symmetrical only with respect to the middle of any and all of its diagonals;
Thirdly, if you take and draw diagonals between all opposite vertices parallelogram, then they will have only one point of intersection.
Fourthly, a square is the length of its diagonal, equal to the sum squares of its 3 dimensions.
Historical reference
Over a period of different historical eras V different countries used various systems measurements of mass, length and other quantities. But since it made it difficult trade relations between countries, and also hampered the development of sciences, there was a need to have a unified international system measures that would be convenient for all countries.
The metric SI system of measures, which suited most countries, was developed in France. Thanks to Mendeleev, the metric system of measures was introduced in Russia.
But many professions to this day use their own specific metrics, sometimes this is a tribute to tradition, sometimes a matter of convenience. For example, sailors still prefer to measure speed in knots, and distance in miles - this is a tradition for them. But jewelers all over the world give preference to such a unit of measurement as the carat - and in their case, this is both tradition and convenience.
Questions:
1. Who knows how many meters are in one mile? What is one node?
2. Why is the unit of measurement for diamonds called “carat”? Why has it historically been convenient for jewelers to measure mass in such units?
3. Who remembers in what units oil is measured?
>> Lesson 31. Formula for the volume of a rectangular parallelepiped
A rectangular parallelepiped is a spatial figure limited rectangles.
Many objects from the environment have a parallelepiped shape: a box, cubes, TV, wardrobe, etc..
Math lesson in 5th grade. (Vilenkin)
Subject: Volumes. Volume of a rectangular parallelepiped.
Target: 1. Consolidate knowledge on this topic when solving problems. Prepare for test work. Give the ratio of volume units.
2. Repeat the properties of multiplication, simplification of expressions, parts of a parallelepiped.
3. Educate environmental aspect, attention.
Equipment: on the board: topic, task for oral counting; Handout: models of parallelepiped, cube, matchbox; for children: cheat sheets, rulers, two-color signal circles,
During the classes.
Organizing time.
Good afternoon, happy hour, math is here. On the desk: rulers, cheat sheets, notebooks, textbooks.
Oral counting (warm-up) No. 806 – in rows “in a chain”,
- apply distributive property multiplication:
(x + 8) 20 on the board
247 123 – 147 123
- simplify:
20a – 19a 4x + x – 2x
13v - 27 + 13v - 10v
Communicate the topic and purpose.
— What geometric figures did you get acquainted with? Today we will repeat how to find the volume of a rectangular parallelepiped and the units of volume. Getting ready for the test.
IV. Repetition of what has been learned. cube models,
— Show top, back, bottom and front edges. parallelepiped
— Show two faces that have a common edge,
— Show vertical edges.
(show 2 or 3 students at the same time)
Game "Yes - no"
— Any cube is a rectangular parallelepiped (+) signal
— A rectangular parallelepiped has 10 vertices (-, 8) circles
– 6 edges (+) – 12 edges (+)
— Each face of the cube is a square (+)
— If the length of a rectangular parallelepiped is not equal to its height, then it cannot be a cube (+)
— The volume of a rectangular parallelepiped is equal to the product of its three dimensions (+)
Find the formula.
- calculate the volume matchbox, cube, parallelepiped. visibility
— additional material“How much air does a person need to breathe?”
With each inhalation, a person introduces 9 liters of air into his lungs in 1 minute. This amounts to 9 * 60 per hour, i.e. 540 liters. Let’s round up to 500 liters or half a cubic meter and find out that a person inhales 12 m³ of air per day. This volume is 14 kg.
In one day, a person passes through his body more air than food: no one eats even 3 kg per day, but we inhale 14 kg. If we consider that the inhaled air consists of 4/5 nitrogen, which is useless for breathing, then it seems that our body consumes only 3 kg, i.e. approximately the same amount as food (solid and liquid).
Do I need any other proof of the need to renew the air in the living room?
- No. 804, 801 - on the board,
— How to calculate the volume of a parallelepiped or cube?
— In what units is volume measured?
VI. Ratio of volume units.“cheat sheets” Write in “cheat sheets”. flyleaf
— Game “The Weakest Link” — No. 802,
— Task on cards.
— Express in cubic cm:
6 dm³, 287 dm³
5 dm³ 23 cm³ 16000 mm³
5 dm³ 635 cm³ 2 dm³ 80 cm³
— Express in cubic dm:
6m³ 580cm³ 7m³ 15dm³
VII. Repetition of what has been learned. № 808
VIII. Result:— What do you remember from the lesson?
— Who worked for 5? by 4?
IX. Homework : § 21, No. 822 (a, b), No. 823.
Mathematics
5th grade
21. Volumes.
If you fill the mold with wet sand, and then turn it over and remove it, you will get figures that have the same volume (Fig. 83). If the mold is filled with water, the volume of water will be equal to volume each sand figure.
Rice. 83
To compare the volumes of two vessels, you can fill one of them with water and pour it into the second vessel. If the second vessel is filled and there is no water left in the first vessel, then the volumes of the vessels are equal. If water remains in the first vessel, then its volume is greater than the volume of the second vessel. And if it is not possible to fill the second vessel with water, then the volume of the first vessel is less than the volume of the second.
The following units are used to measure volumes: cubic millimeter (mm3), cubic centimeter (cm3), cubic decimeter (dm3), cubic meter (m3), cubic kilometer (km3).
For example: a cubic centimeter is the volume of a cube with an edge of 1 cm (Fig. 84).
Rice. 84
A cubic decimeter is also called a liter.
The figure in Figure 85 consists of 4 cubes with an edge of 1 cm. This means that its volume is 4 cm3.
Rice. 85
Let us derive a rule for calculating the volume of a rectangular parallelepiped.
Formulas for the volumes of parallelepipeds and cubes
Let a rectangular parallelepiped have a length of 4 cm, a width of 3 cm and a height of 2 cm (Fig. 86, a). Let's divide it into two layers 1 cm thick (Fig. 86, b). Each of these layers consists of 3 columns 4 cm long (Fig. 86, c), and each column consists of 4 cubes with an edge of 1 cm (Fig. 86, d). This means that the volume of each column is 4 cm3, each layer is 4 3 (cm3), and the entire rectangular parallelepiped is (4 3) 2, that is, 24 cm3.
Rice. 86
To find the volume of a rectangular parallelepiped, you need to multiply its length by its width and height.
The formula for the volume of a rectangular parallelepiped is
where V is volume; a, b, c - measurements.
If the edge of a cube is 4 cm, then the volume of the cube is 4 4 4 = 43 (cm3), that is, 64 cm3.
If the edge of a cube is equal to a, then the volume V of the cube is equal to a a a = a3.
This means that the formula for the volume of a cube has the form
That is why the entry a3 is called the cube of a.
The volume of a cube with an edge of 1 m is equal to 1 m3. And since 1 m = 10 dm, then 1 m3 = 103 dm3, that is, 1 m3 = 1000 dm3 = 1000 l.
In the same way we find that
1 l = 1 dm3 = 1000 cm3; 1 cm3 = 1000 mm3;
1 km3 = 1,000,000,000 m3 (see figure).
Self-test questions
- The figure consists of 19 cubes with a side of 1 cm each; what is the volume of this figure?
- What is a cubic centimeter; cubic meter?
- What is another name for cubic decimeter?
- How many cubic centimeters is 1 liter?
- How many liters is a cubic meter equal?
- How many cubic meters in cubic kilometers?
- Write the formula for the volume of a rectangular parallelepiped.
- What does the letter V mean in this formula; letters a, b, c?
- Write the formula for the volume of a cube.
Do the exercises
819. Figures are made from cubes with an edge of 1 cm (Fig. 87). Find the volumes and surface areas of these figures.
Rice. 87
820. Find the volume of a rectangular parallelepiped if:
- a) a = 6 cm, b = 10 cm, c = 5 cm;
- b) a = 30 dm, b = 20 dm, c = 30 dm;
- c) a = 8 dm, b = 6 m, c = 12 m;
- d) a = 2 dm 1 cm, b = 1 dm 7 cm, c = 8 cm;
- e) a = 3 m, b = 2 dm, c = 15 cm.
821. Square bottom edge of a rectangular parallelepiped is 24 cm2. Determine the height of this parallelepiped if its volume is 96 cm3.
822. The volume of the room is 60 m3. The height of the room is 3 m, the width is 4 m. Find the length of the room and the area of the floor, ceiling, and walls.
823. Find the volume of a cube whose edge is 8 dm; 3 dm 6 cm.
824. Find the volume of a cube if its surface area is 96 cm2.
825. Express:
- a) in cubic centimeters: 5 dm3 635 cm3; 2 dm3 80 cm3;
- b) in cubic decimeters: 6 m3 580 dm3; 7 m3 15 dm3;
- c) in cubic meters and decimeters: 3270 dm3; 12,540,000 cm3.
826. The height of the room is 3 m, width 5 m and length 6 m. How many cubic meters of air are in the room?
827. The length of the aquarium is 80 cm, the width is 45 cm, and the height is 55 cm. How many liters of water must be poured into this aquarium so that the water level is 10 cm below the top edge of the aquarium?
828. The rectangular parallelepiped (Fig. 88) is divided into two parts. Find the volume and surface area of the entire parallelepiped and both of its parts. Is the volume of a parallelepiped equal to the sum of the volumes of its parts? Can this be said about their surface areas? Explain why.
Rice. 88
829. Calculate orally:
830. Restore the chain of calculations:
831. Find the meaning of the expression:
- a) 23 + Z2;
- b) 33 + 52;
- c) 43 + 6;
- d) 103 - 10.
832. How many tens are there in the quotient:
- a) 1652: 7;
- b) 774: 6;
- c) 1632: 12;
- d) 2105: 5?
833. Do you agree with the statement:
- a) any cube is also a rectangular parallelepiped;
- b) if the length of a rectangular parallelepiped is not equal to its height, then it cannot be a cube;
- c) each face of a cube is a square?
834. Four identical barrels hold 26 buckets of water. How many buckets of water can 10 of these barrels hold?
835. In how many ways from 7 beads different colors can you make a necklace (with a clasp)?
836. Name in a rectangular parallelepiped (Fig. 89):
- a) two faces having a common edge;
- b) top, back, front and bottom edges;
- c) vertical ribs.
Rice. 89
837. Solve the problem:
- Find the area of each plot if the area of the first plot is 5 times more area the second, and the area of the second is 252 hectares less area first.
- Find the area of each plot if the area of the second plot is 324 hectares greater than the area of the first plot, and the area of the first plot is 7 times less than the area of the second.
838. Follow these steps:
- 668 (3076 + 5081);
- 783 (66 161 — 65 752);
- 2 111 022: (5960 — 5646);
- 2 045 639: (6700 — 6279).
839. In Rus', in the old days, a bucket (about 12 liters), a shtof (a tenth of a bucket) was used as units of volume measurement; in the USA, England and other countries a barrel (about 159 liters), a gallon (about 4 liters), a bushel (about 36 l), pint (from 470 to 568 cubic centimeters). Compare these units. Which ones are larger than 1 m3?
840. Find the volumes of the figures shown in Figure 90. The volume of each cube is 1 cm3.
Rice. 90
841. Find the volume of a rectangular parallelepiped (Fig. 91).
Rice. 91
842. Find the volume of a rectangular parallelepiped if its dimensions are 48 dm, 16 dm and 12 dm.
843. The barn, shaped like a rectangular parallelepiped, is filled with hay. The length of the barn is 10 m, width 6 m, height 4 m. Find the mass of hay in the barn if the mass of 10 m3 of hay is 6 quintals.
844. Express in cubic decimeters:
- 2 m3 350 dm3;
- 3 m3 7 dm3;
- 4 m3 30 dm3;
- 18,000 cm3;
- 210,000 cm3.
845. The volume of a rectangular parallelepiped is 1248 cm3. Its length is 13 cm and its width is 8 cm. Find the height of this parallelepiped.
846. Using the formula V = abc calculate:
- a) V, if a - 3 dm, b = 4 dm, c = 5 dm;
- b) a, if V = 2184 cm3, b = 12 cm, c = 13 cm;
- c) b, if V = 9200 cm3, a = 23 cm, c = 25 cm;
- d) ab, if V = 1088 dm3, c = 17 cm.
What is the meaning of ab?
847. Father older than my son for 21 years. Write down a formula expressing - the age of the father - through b - the age of the son. Find using this formula:
- a) a, if b = 10;
- b) a, if b = 18;
- c) b, if a = 48.
848. Find the meaning of the expression:
- a) 700,700 - 6054 (47,923 - 47,884) - 65,548;
- b) 66,509 + 141,400: (39,839 - 39,739) + 1985;
- c) (851 + 2331) : 74 - 34;
- d) (14,084: 28 - 23) 27 - 12,060;
- e) (102 + 112 + 122) : 73 + 895;
- f) 2555: (132 + 142) + 35.
849. Calculate from the table (Fig. 92):
- a) how many times does the number 9 appear;
- b) how many times do the numbers 6 and 7 appear in the table (not counting them separately);
- c) how many times do the numbers 5, 6 and 8 appear (not counting them individually).
Rice. 92
Stories about the history of the emergence and development of mathematics
200 years ago in different countries, including Russia, different systems of units were used to measure length, mass and other quantities. The relationships between the measures were complex, there were different definitions for units of measurement.
For example, to this day in Great Britain there are two different “tons” (2000 and 2940 pounds), more than 50 different “bushels”, etc. This hampered the development of science and trade between countries, so there is a need to introduce a unified system of measures , convenient for all countries, with simple relationships between units.
Such a system - it was called the metric system of measures - was developed in France. Basic unit of length, 1 meter (from Greek word“metron” - measure), defined as a forty-millionth fraction of the Earth’s circumference, the basic unit of mass, 1 kilogram - as the mass of 1 dm3 clean water. The remaining units were determined through these two, the ratios between units of the same value were equal to 10, 100, 1000, etc.
The metric system of measures has been adopted by most countries of the world; in Russia its introduction began in 1899. Great contribution to the introduction and dissemination metric system measures in our country belong to Dmitry Ivanovich Mendeleev, the great Russian chemist.
However, according to tradition, even today the old units are sometimes used. sailors measure distances in miles (1852 m) and cables (a tenth of a mile, that is, about 185 m), speed - in knots (1 mile per hour). The mass of diamonds is measured in carats (200 mg, that is, a fifth of a gram is the mass of a wheat grain). The volume of oil is measured in barrels (159 l), etc.
This can be done different ways, it all depends on what quantities and objects we have.
So, the first method, which is suitable exclusively for a rectangular parallelepiped.
To determine the volume of a parallelepiped you will need its height, width and length.
Since rectangles form a parallelepiped, let's mark their length and width with the letters a and b, respectively. Then the area of the rectangle will be calculated as a*b.
The height of a parallelepiped is the height lateral rib, and since the height is a constant value, to find the volume you need to multiply the base area of the parallelepiped by the height. This is expressed by the following formula: V = a*b*c = S*c, where c is the height.
Let's look at an example. Let's say we have a parallelepiped with a base length and width of 5 and 8 cm, and its height is 11 cm. It is necessary to calculate the volume.
Find the area of the base: 5*8=40 sq. cm. Now we multiply the resulting value by the height 40*11=440 cubic meters. cm is the volume of the figure.
Second way.
Since the base of the parallelepiped is geometric figure parallelogram, you need to determine its area. To find the area of a parallelogram depending on the known data, you can use the following formulas:
- S = a*h, where a is the side of the parallelogram, h is the height drawn to a.
- S = a*b*sinα, where a and b are the sides of the figure, α is the angle between these sides.
After that. How did you figure it out? How to find the area of a parallelogram, you can begin to find the volume of our parallelepiped. To do this we use the formula:
V = S*h, where S is the base area obtained earlier, h is the height of our parallelepiped.
Let's look at an example.
We are given a parallelepiped with a height of 50 cm, the base (parallelogram) of which has a side equal to 23 cm and the height drawn to this side is 8 cm. We substitute the above formula:
S = 23*8 = 184 sq. cm.
Now we substitute the formula to find the volume of a parallelepiped:
V = 184*50 = 9,200 cubic meters
Mathematics lesson ‘Volume of a rectangular parallelepiped’ (5th grade)
Answer: the volume of this parallelepiped is 9200 cubic centimeters.
Third way.
This option is only suitable for rectangular type parallelepiped, sides whose bases will be equal. To do this, you just need to cube these sides.
V = a3, i.e. cubed
Given a parallelepiped with a base side of 12. This means that the volume of this figure is calculated by the following formula V = 123 = 1728 cc cm.
Either method is very simple. The main thing is to arm yourself with a calculator and perform all the calculations correctly. Good luck!
volume of a rectangular parallelepiped
S1*2 + S2*2 + S3*2 = S
Parallelepiped base
The calculator will calculate and write out the solution in detail and with comments. All you have to do is copy the line solution of the parallelepiped into your notebook. A detailed text solution with explanations will allow you to understand the methodology for solving such problems and, if necessary, answer questions by giving a detailed and competent answer.
Calculation of volume and area of a parallelogram is an elementary basis for many technical and everyday calculations!
Volumes. Volume of a rectangular parallelepiped
For example, to calculate repairs in a room, calculate data for heating or air conditioning.
rectangular parallelogram
The formula used in our calculator will find volume of a rectangular parallelepiped. And if your parallelepiped has oblique edges, instead of the length of the corresponding oblique edge, you must enter the value of the height of this part of the figure.
Formula for the volume of a rectangular parallelepiped
To find it, you need to know the dimensions of the ribs: height, width and length. According to the formula, the dimensions of the parallelepiped faces must be multiplied in any order.
The volume can be expressed in liters or cubic cm, cubic millimeters.
Formula for the surface area of a parallelepiped
S1*2 + S2*2 + S3*2 = S
Using the formula for the area of a parallelepiped, you need to find the areas of all sides of the parallelepiped and then add them up. Opposite sides, faces, and edges of a parallelepiped are equal to each other, so when calculating areas you can use multiplication by two.
Parallelepiped base
In some cases, the base area of the parallelepiped is known, then in order to find the volume it is enough to multiply the base area by the height. ! IMPORTANT! - this is true only for a rectangular parallelepiped.
How to find the volume of a parallelepiped?
The easiest way to find the volume is by entering three known values into columns online calculator volume! Then - press the button - you will get the result)!
The calculator will calculate volume of parallelepiped abcda1b1c1d1 and will describe the decision in detail and with comments.
Volume of a rectangular parallelepiped
All you have to do is copy the line solution of the parallelepiped into your notebook. A detailed text solution with explanations will allow you to understand the methodology for solving such problems and, if necessary, answer questions by giving a detailed and competent answer.
Calculation of volume and area of a parallelogram is an elementary basis for many technical and everyday calculations! For example, to calculate repairs in a room, calculate data for heating or air conditioning.
A parallelogram is a three-dimensional geometric figure that has six sides, each side being a parallelogram. The sides of a parallelogram are usually called faces. If all the faces of a parallelepiped have the shape of a rectangle, then this is already rectangular parallelogram! This figure is designated by the letters abcda1b1c1d1.
