The base area of ​​a regular hexagonal prism is 3. The largest diagonal of a regular hexagonal prism, having length d, makes an angle α with the lateral edge of the prism

In a triangle B E B1 :

• B B1 = h
• B E = 2 ⋅ a- because E O = O B = a
• ∠ E B B1 = 90 ∘ - according to the properties of the correct straightness

Thus, it turns out that the triangle B E B1 rectangular. According to the properties of a right triangle

E B1 = B B2 1 +B E2 − − − − − − − − − − √ = h2 + 4 ⋅ a2 − − − − − − − − √

If h = a, so then

E B1 = 5 √ ⋅ a

After similar reasoning we obtain that F C1 = A D1 = B E1 = C F1 = D A1 = h2 + 4 ⋅ a2 − − − − − − − − √ .

We find O F1

In a triangle F O F1 :

• F F1 = h
• F O = a
• ∠ O F F1 = 90 ∘ - according to the properties of a regular prism

Thus, it turns out that the triangle F O F1 rectangular. According to the properties of a right triangle

O F1 = F F2 1 + O F2 − − − − − − − − − − √ = h2 + a2 − − − − − − √

If h = a, so then

The site has already discussed some types of problems in stereometry, which are included in a single bank of tasks for the mathematics exam.For example, tasks about .

A prism is called regular if its sides are perpendicular to the bases and a regular polygon lies at the bases. That is, a regular prism is a straight prism with a regular polygon at its base.

A regular hexagonal prism has a regular hexagon at the base, the side faces are rectangles.

In this article you will find problems to solve a prism, the base of which is a regular hexagon. There are no special features or difficulties in the solution. What's the point? Given a regular hexagonal prism, you need to calculate the distance between two vertices or find a given angle. The problems are actually simple; in the end, the solution comes down to finding an element in a right triangle.

The Pythagorean theorem is used and. Knowledge of the definitions of trigonometric functions in a right triangle is required.

Be sure to look at the information about the regular hexagon in.You will also need the skill of extracting a large number of them. You can solve polyhedra, they also calculated the distance between vertices and angles.

Briefly: what is a regular hexagon?

It is known that in a regular hexagon the sides are equal. In addition, the angles between the sides are also equal.

*Opposite sides are parallel.

Additional Information

The radius of a circle circumscribed about a regular hexagon is equal to its side. *This is confirmed very simply: if we connect the opposite vertices of a hexagon, we get six equal equilateral triangles. Why equilateral?

Each triangle has an angle with its vertex lying in the center equal to 60 0 (360:6=60). Since the two sides of a triangle having a common vertex in the center are equal (these are the radii of the circumscribed circle), then each angle at the base of such an isosceles triangle is also equal to 60 degrees.

That is, a regular hexagon, figuratively speaking, consists of six equal equilateral triangles.

What other fact should be noted that is useful for solving problems? The vertex angle of a hexagon (the angle between its adjacent sides) is 120 degrees.

*We deliberately did not touch upon the formulas for a regular N-gon. We will consider these formulas in detail in the future; they are simply not needed here.

Let's consider the tasks:

272533. In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are equal 48. Find the distance between points A and E 1 .

Consider right triangle AA 1 E 1 . According to the Pythagorean theorem:

*The angle between the sides of a regular hexagon is 120 degrees.

Section AE 1 is the hypotenuse, AA 1 and A 1 E 1 legs. Rib AA 1 we know. Catet A 1 E 1 we can find using using .

Theorem: The square of any side of a triangle is equal to the sum of the squares of its two other sides without twice the product of these sides by the cosine of the angle between them.

Hence

According to the Pythagorean theorem:

Answer: 96

*Please note that squaring 48 is not necessary.

In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are 35. Find the distance between points B and E.

It is said that all edges are equal to 35, that is, the side of the hexagon lying at the base is equal to 35. And also, as already said, the radius of the circle described around it is equal to the same number.

Thus,

Answer: 70

273353. In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are equal to forty roots of five. Find the distance between points B and E 1.

Consider right triangle BB 1 E 1 . According to the Pythagorean theorem:

Segment B 1 E 1 is equal to two radii of the circle circumscribed about a regular hexagon, and its radius is equal to the side of the hexagon, that is

Thus,

Answer: 200

273683. In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are equal to 45. Find the tangent of the angle AD 1 D.

Consider a right triangle ADD 1 in which AD equal to the diameter of a circle circumscribed around the base. It is known that the radius of a circle circumscribed around a regular hexagon is equal to its side.

Thus,

Answer: 2

In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are equal 23. Find the angle DAB. Give your answer in degrees.

Consider a regular hexagon:

In it, the angles between the sides are 120°. Means,

The length of the edge itself does not matter; it does not affect the angle.

Answer: 60

In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are equal to 10. Find the angle AC 1 C. Give the answer in degrees.

Consider the right triangle AC 1 C:

Let's find A.C.. In a regular hexagon, the angles between its sides are equal to 120 degrees, then according to the cosine theorem for a triangleABC:

Thus,

So angle AC 1 C is equal to 60 degrees.

Answer: 60

274453. In a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 all edges are equal to 10. Find the angle AC 1 C. Give the answer in degrees.

From each vertex of a prism, for example from vertex A 1 (Fig.), three diagonals can be drawn (A 1 E, A 1 D, A 1 C).

They are projected onto the plane ABCDEF by the diagonals of the base (AE, AD, AC). Of the inclined ones A 1 E, A 1 D, A 1 C, the largest is the one with the largest projection. Consequently, the largest of the three diagonals taken is A 1 D (in the prism there are also diagonals equal to A 1 D, but there are no larger ones).

From triangle A 1 AD, where ∠DA 1 A = α and A 1 D = d , we find H=AA 1 = d cos α ,
AD= d sin α .

The area of ​​an equilateral triangle AOB is equal to 1/4 AO 2 √3. Hence,

S ocn. = 6 1/4 AO 2 √3 = 6 1/4 (AD/2) 2 √3.

Volume V = S H = 3√ 3 / 8 AD 2 AA 1

Answer: 3√ 3 / 8 d 3 sin 2 α cos α .

Comment . To depict a regular hexagon (the base of a prism), you can construct an arbitrary parallelogram BCDO. Laying out the segments OA = OD, OF = OC and OE = OB on the continuations of lines DO, CO, BO, we obtain the hexagon ABCDEF. Point O represents the center.

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 ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; 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. From a physical point of view, this 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 at a 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? Remain in constant units of time and do not switch to reciprocal units. 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 this is not a complete solution to the problem. 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, the logical paradox is 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 a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to 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.” Reasonable beings will never understand such absurd logic. 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 “mind 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. Let us apply mathematical set theory to 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.” Let us 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: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...

And now I have the most interesting question: 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 the digits of a 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 this is 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 number systems 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 the large number 12345, I don’t want to fool my head, let’s consider 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 a fool who doesn’t know physics. She just has a strong stereotype of perceiving 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.