Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. For 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 paces 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 insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:
A flying arrow is motionless, since at each 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, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that is inextricably linked to reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: on different coins there is different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can 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 elementarily.
Let's figure out what and how we do in order to find the sum of digits given number. And so, let's say we 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 to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up 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" from shamans used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, 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. FROM a large number 12345 I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of 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 action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I don't think that girl is stupid, no who knows physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
Often students indignantly ask: “How will this be useful to me in life?”. On any topic of each subject. The topic about the volume of a parallelepiped is no exception. And here it is just possible to say: "It will come in handy."
How, for example, to find out if a parcel will fit in a mailbox? Of course, you can choose the right one by trial and error. What if there is no such possibility? Then calculations will come to the rescue. Knowing the capacity of the box, you can calculate the volume of the parcel (at least approximately) and answer the question.
Parallelepiped and its types
If we literally translate its name from ancient Greek, it turns out that this is a figure consisting of parallel planes. There are such equivalent definitions of a parallelepiped:
- a prism with a base in the form of a parallelogram;
- polyhedron, each face of which is a parallelogram.
Its types are distinguished depending on which figure lies at its base and how the side ribs are directed. In general, one speaks of oblique parallelepiped whose base and all faces are parallelograms. If the side faces of the previous view become rectangles, then it will need to be called already direct. And at rectangular and the base also has 90º angles.
Moreover, in geometry they try to depict the latter in such a way that it is noticeable that all the edges are parallel. Here, by the way, the main difference between mathematicians and artists is observed. It is important for the latter to convey the body in compliance with the law of perspective. And in this case, the parallelism of the edges is completely invisible.
About the introduced notation
In the formulas below, the designations indicated in the table are valid.
Formulas for an oblique box
The first and second for areas:
The third one is for calculating the volume of the box:
Since the base is a parallelogram, to calculate its area, you will need to use the appropriate expressions.
Formulas for a cuboid
Similarly to the first paragraph - two formulas for areas:
And one more for volume:
First task
Condition. Given a rectangular parallelepiped whose volume is to be found. The diagonal is known - 18 cm - and the fact that it forms angles of 30 and 45 degrees with the plane of the side face and the side edge, respectively.
Solution. To answer the question of the problem, you need to find out all the sides in three right triangles. They will give the necessary edge values for which you need to calculate the volume.
First you need to figure out where the 30º angle is. To do this, you need to draw a diagonal of the side face from the same vertex from which the main diagonal of the parallelogram was drawn. The angle between them will be what you need.
The first triangle, which will give one of the sides of the base, will be the following. It contains the desired side and two diagonals drawn. It is rectangular. Now you need to use the ratio of the opposite leg (base side) and the hypotenuse (diagonal). It is equal to the sine of 30º. That is, the unknown side of the base will be determined as the diagonal multiplied by the sine of 30º or ½. Let it be marked with the letter "a".
The second will be a triangle containing a known diagonal and an edge with which it forms 45º. It is also rectangular, and you can again use the ratio of the leg to the hypotenuse. In other words, the side edge to the diagonal. It is equal to the cosine of 45º. That is, "c" is calculated as the product of the diagonal and the cosine of 45º.
c = 18 * 1/√2 = 9 √2 (cm).
In the same triangle, you need to find another leg. This is necessary in order to then calculate the third unknown - "in". Let it be marked with the letter "x". It is easy to calculate using the Pythagorean theorem:
x \u003d √ (18 2 - (9 √ 2) 2) \u003d 9 √ 2 (cm).
Now we need to consider another right triangle. It already contains famous parties"s", "x" and the one that needs to be counted, "in":
c \u003d √ ((9 √ 2) 2 - 9 2 \u003d 9 (cm).
All three quantities are known. You can use the formula for volume and calculate it:
V \u003d 9 * 9 * 9√2 \u003d 729√2 (cm 3).
Answer: the volume of the parallelepiped is 729√2 cm 3 .
Second task
Condition. Find the volume of the parallelepiped. It knows the sides of the parallelogram that lies at the base, 3 and 6 cm, as well as its acute angle - 45º. The lateral rib has an inclination to the base of 30º and is equal to 4 cm.
Solution. To answer the question of the problem, you need to take the formula that was written for the volume of an inclined parallelepiped. But both quantities are unknown in it.
The area of \u200b\u200bthe base, that is, the parallelogram, will be determined by the formula in which you need to multiply the known sides and the sine of the acute angle between them.
S o \u003d 3 * 6 sin 45º \u003d 18 * (√2) / 2 \u003d 9 √2 (cm 2).
The second unknown is the height. It can be drawn from any of the four vertices above the base. It can be found from a right triangle, in which the height is the leg, and the side edge is the hypotenuse. In this case, an angle of 30º lies opposite the unknown height. So, you can use the ratio of the leg to the hypotenuse.
n \u003d 4 * sin 30º \u003d 4 * 1/2 \u003d 2.
Now all values are known and you can calculate the volume:
V \u003d 9 √2 * 2 \u003d 18 √2 (cm 3).
Answer: the volume is 18 √2 cm 3 .
Third task
Condition. Find the volume of the parallelepiped if it is known to be a straight line. The sides of its base form a parallelogram and are equal to 2 and 3 cm. Sharp corner between them 60º. The smaller diagonal of the parallelepiped is equal to the larger diagonal of the base.
Solution. In order to find out the volume of a parallelepiped, we use the formula with the base area and height. Both quantities are unknown, but they are easy to calculate. The first one is height.
Since the smaller diagonal of the parallelepiped is the same size as the larger base, they can be denoted by the same letter d. The largest angle of a parallelogram is 120º, since it forms 180º with an acute one. Let the second diagonal of the base be denoted by the letter "x". Now, for the two diagonals of the base, cosine theorems can be written:
d 2 \u003d a 2 + in 2 - 2av cos 120º,
x 2 \u003d a 2 + in 2 - 2ab cos 60º.
Finding values without squares does not make sense, since then they will be raised to the second power again. After substituting the data, it turns out:
d 2 \u003d 2 2 + 3 2 - 2 * 2 * 3 cos 120º \u003d 4 + 9 + 12 * ½ \u003d 19,
x 2 \u003d a 2 + in 2 - 2ab cos 60º \u003d 4 + 9 - 12 * ½ \u003d 7.
Now the height, which is also the side edge of the parallelepiped, will be the leg in the triangle. The hypotenuse will be known diagonal body, and the second leg - "x". You can write the Pythagorean Theorem:
n 2 \u003d d 2 - x 2 \u003d 19 - 7 \u003d 12.
Hence: n = √12 = 2√3 (cm).
Now the second unknown quantity is the area of the base. It can be calculated using the formula mentioned in the second problem.
S o \u003d 2 * 3 sin 60º \u003d 6 * √3/2 \u003d 3 √3 (cm 2).
Combining everything into a volume formula, we get:
V = 3√3 * 2√3 = 18 (cm 3).
Answer: V \u003d 18 cm 3.
The fourth task
Condition. It is required to find out the volume of a parallelepiped that meets the following conditions: the base is a square with a side of 5 cm; side faces are rhombuses; one of the vertices above the base is equidistant from all the vertices lying at the base.
Solution. First you need to deal with the condition. There are no questions with the first paragraph about the square. The second, about rhombuses, makes it clear that the parallelepiped is inclined. Moreover, all its edges are equal to 5 cm, since the sides of the rhombus are the same. And from the third it becomes clear that the three diagonals drawn from it are equal. These are two that lie on the side faces, and the last one is inside the parallelepiped. And these diagonals are equal to the edge, that is, they also have a length of 5 cm.
To determine the volume, you will need a formula written for an inclined parallelepiped. Again, there are no known quantities in it. However, the area of the base is easy to calculate because it is a square.
S o \u003d 5 2 \u003d 25 (cm 2).
A little more difficult is the case with height. It will be such in three figures: a parallelepiped, a quadrangular pyramid and an isosceles triangle. The last circumstance should be used.
Since it is a height, it is a leg in right triangle. The hypotenuse in it will be a known edge, and the second leg is equal to half the diagonal of the square (the height is also the median). And the diagonal of the base is easy to find:
d = √(2 * 5 2) = 5√2 (cm).
The height will need to be calculated as the difference of the second degree of the edge and the square of half the diagonal and do not forget to extract the square root:
n = √ (5 2 - (5/2 * √2) 2) = √(25 - 25/2) = √(25/2) = 2.5 √2 (cm).
V \u003d 25 * 2.5 √2 \u003d 62.5 √2 (cm 3).
Answer: 62.5 √2 (cm 3).
Volume of the box
The volume value 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 the height.
In everyday life, most often, to measure the volume of liquid, as a rule, they use such a measuring unit as liter = 1dm3.
In addition to this unit of measurement, the following are used to determine the volume:
The parallelepiped belongs to the simplest three-dimensional figures and therefore it is not difficult to find its volume.
Volume of the box is 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 box
And now let's remember what a parallelepiped is and how it differs from a cube.
A parallelepiped is a three-dimensional figure, at the base of which lies a polygon. The surface of a cuboid consists of six rectangles, which are the faces of this cuboid. 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 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 picture and tell me what it reminds you of?
2. Think and give an answer, where in everyday life can you encounter such a figure?
3. How many edges does the parallelepiped have?
Varieties of parallelepipeds
Parallelepipeds are divided into several varieties, such as:
Rectangular;
Inclined;
cube.
Rectangular parallelepipeds include those figures whose faces consist of rectangles.
If the side faces are not perpendicular to its base, then you have an inclined parallelepiped.
A figure such as a cube is also a parallelepiped. Without exception, all its faces are in the form of squares.
Box properties
The figure under study has a number of properties, which we will now learn about:
First, the opposite faces of this figure are equal and parallel to each other;
Secondly, it is symmetrical only with respect to the middle of any of its diagonals without exception;
Thirdly, if you take and draw diagonals between all opposite vertices of a parallelogram, then they will have only one intersection point.
Fourth, the square is the length of its diagonal, is equal to the sum squares of its 3 dimensions.
History reference
For a period of different historical eras in different countries used various systems for measuring mass, length and other quantities. But since this hampered trade relations between countries, and also hampered the development of science, it became necessary to have a unified international system of measures that would be convenient for all countries.
The metric SI system, which suited most countries, was developed in France. Thanks to Mendeleev, the metric system of measures was also introduced in Russia.
But many professions still use their own specific metrics, sometimes it's a tribute to tradition, sometimes it's a matter of convenience. So, for example, sailors still prefer to measure speed in knots, and distance in miles is a tradition for them. But jewelers around the world prefer such a unit of measure as carat - and in their case it 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 is it historically convenient for jewelers to measure mass in such units?
3. Who remembers the units in which oil is measured?
Rectangle- one of the simplest flat figures, and a rectangular parallelepiped is the same simple figure, but in space (Fig. 1). They are very similar.
As similar as a circle and a sphere.
Rice. 1. Rectangle and box
A conversation about areas begins with the area of a rectangle, and about volumes - with the volume of a rectangular parallelepiped.
If we know how to find the area of a rectangle, then this allows us to find the area of any figure.
We can divide this figure into 3 rectangles and find the area of each, and hence the entire figure. (Fig. 2.)
Rice. 2. Figure
Rice. 3. A figure whose area is equal to seven rectangles
Even if the figure is not divided exactly into rectangles, this can be done with any accuracy and the area can be calculated approximately.
The area of this figure (Fig. 3) is approximately equal to the sum of the areas of seven rectangles. The inaccuracy is obtained due to the upper small figures. If you increase the number of rectangles, then the inaccuracy will decrease.
That is rectangle is a tool for calculating the area of any figure.
The situation is the same when it comes to volumes.
Any figure can be laid out with rectangular parallelepipeds, bricks. The smaller these bricks are, the more accurately we can calculate the volume (Fig. 4, Fig. 5).
Rice. 4. Calculating the area using rectangular parallelepipeds
The cuboid is a tool for calculating the volumes of any figure.
Rice. 5. Calculate the area using small boxes
Let's remember a little.
A square with a side of 1 unit (Fig. 6) has an area of 1 square unit. The initial linear unit can be any: centimeter, meter, kilometer, mile.
For example, 1 cm2 is the area of a square with a side of 1 cm.
Rice. 6. Square and rectangle
Rectangle area is the number of such squares that will fit in it. (Fig. 6.)
We lay the unit squares in the length of the rectangle in one row. Got 5 pieces.
3 squares are placed in height. This means that there are three rows in total, each with five squares.
The total area is .
It is clear that there is no need to place unit squares inside the rectangle each time.
It is enough to multiply the length of one side by the length of the other.
Or in general view:
The situation is very similar with the volume of a rectangular parallelepiped.
The volume of a cube with a side of 1 unit is 1 cubic unit. Again, the original linear values can be anything: millimeters, centimeters, inches.
For example, 1 cm 3 is the volume of a cube with a side of 1 cm, and 1 km 3 is the volume of a cube with a side of 1 km.
Let's find the volume of a rectangular parallelepiped with sides 7 cm, 5 cm, 4 cm. (Fig. 7.)
Rice. 7. Rectangular box
The volume of our cuboid is the number of unit cubes that fit into it.
Lay on the bottom a row of single cubes with a side of 1 cm along the long side. Fitted 7 pieces. Already from experience with a rectangle, we know that only 5 such rows will fit on the bottom, 7 pieces in each. That is all:
Let's call it a layer. How many such layers can we stack on top of each other?
It depends on the height. It is equal to 4 cm. This means that 4 layers are laid in each of 35 pieces. Total:
Where did the number 35 come from? This is 75. That is, we got the number of cubes by multiplying the lengths of all three sides.
But this is the volume of our rectangular parallelepiped.
Answer: 140
Now we can write the formula in general form. (Fig. 8.)
Rice. 8. Volume of a parallelepiped
The volume of a rectangular parallelepiped with sides , , is equal to the product of all three sides.
If the lengths of the sides are given in centimeters, then the volume will be in cubic centimeters (cm 3).
If in meters, then the volume is in cubic meters (m 3).
Similarly, volume can be measured in cubic millimeters, kilometers, etc.
A glass cube with a side of 1 m is completely filled with water. What is the mass of water? (Fig. 9.)
Rice. 9. Cube
The cube is singular. Side - 1 m. Volume - 1 m 3.
If we know how much 1 cubic meter of water weighs (abbreviated as a cubic meter), then the problem is solved.
But if we do not know this, then it is not difficult to calculate.
Side length.
Let's calculate the volume in dm 3.
But 1 dm 3 has a separate name, 1 liter. That is, we have 1000 liters of water.
We all know that the mass of one liter of water is 1 kg. That is, we have 1000 kg of water, or 1 ton.
It is clear that such a cube filled with water cannot be moved by any ordinary person.
Answer: 1 t.
Rice. 10. Refrigerator
The refrigerator has a height of 2 meters, a width of 60 cm and a depth of 50 cm. Find its volume.
Before we use the volume formula - the product of the lengths of all sides - it is necessary to convert the lengths into the same units of measurement.
We can convert everything to centimeters.
Accordingly, we will get the volume in cubic centimeters.
I think you will agree that volume is more understandable in cubic meters.
A person's eye can't tell a number with five zeros from a number with six zeros, but one is 10 times larger than the other.
Often we need to convert one unit of volume to another. For example, cubic meters to cubic decimetres. It's hard to remember all these ratios. But this does not need to be done. It is enough to understand the general principle.
For example, how many cubic centimeters are in a cubic meter?
Let's see how many cubes with a side of 1 centimeter fit in a cube with a side of 1 m. (Fig. 11.)
Rice. 11. Cube
100 pieces fit in one row (after all, there are 100 cm in one meter).
100 rows or cubes fit into one layer.
There are 100 layers in total.
In this way,
That is, if the linear quantities are related by the ratio “100 cm in one meter”, then in order to get the ratio for cubic quantities, you need to raise 100 to the 3rd power (). And you don't need to draw cubes every time.
A rectangular parallelepiped is a figure, at the base of which there is a rectangle. The figure has six sides. The faces, intersecting, form edges, there are 12 of them.
A rectangular parallelepiped has four side faces. In life, we often encounter this figure: a wardrobe, a refrigerator, a box - they all have the shape of a rectangular parallelepiped.
Rice. 1. Rectangular box
The formula for the volume of this figure
The volume of a cube (a figure with a square at its base) with a side of 1 unit is called 1 cubic unit.
Rice. 2. Unit cube
If the bottom, in order to lay the bottom of the figure with such cubes, will need 4 cubes in length, and 3 in width.
Rice. 3. A rectangular box filled with a ball of cubes
Thus, to fill the base, you must:
3 x 4 \u003d 12 - so we calculated the area.
To fill the entire figure and find out the volume, you need to calculate how many such layers of cubes will fit in the height, for example, if it is 2, then the volume will be:
3 x 4 x 2 = 24 cubes
So, if we take into account that the length of the base of the figure is 4 units, the width is 3, the height is 2, then in order to subtract the volume of a rectangular parallelepiped, it is necessary to find the product of these quantities or measurements. A figure that has three dimensions is called three-dimensional or three-dimensional.
The letter V is used to denote volume.
The formula for the volume of a rectangular parallelepiped is:
$$V = a b c$$
If necessary, all data in the task must be converted to the same unit of measure.
The units are $mm^3, cm^3, dm^3$ and so on. It is important to read correctly: $1 m^3$ and so on.
The English illusionist spent 44 days in a glass cuboid suspended over the River Thames. All he had at his disposal was water, a pillow, a mattress and writing materials.
Exercise: Subtract the volume of a figure whose width is 4 inches, length 50 mm, and height 10 cm.
Solution: First you need to convert all data into one unit of measure.
$4 dm. = 40 cm$;
$50 mm. = 5 cm$.
$V = 40 5 10 = 200 cm^3$
Thus, the volume of the figure is $V = 200 cm^3$
To measure the volume of a liquid, a special unit of measurement is a liter - 1 liter.
Ancient liquid measurements, for example cor = 220 l, baht = 22 l.
Volume measurements:
$$1 l = 1000 cm^3 = 1 dm^3$$
$$1 km^3 = 1000,000,000 m^3$$
$$1 m^3 = 1,000 dm^3 = 1,000,000 cm^3$$
$$1 dm^3 = 1,000 cm^3$$
