One of the most interesting problems of geometry, the result of the decision which is important in physics and chemistry, and other fields - to determine the volume.Catching mathematics at school, children are often given by the thought: "Why do we need it?" The world around seems so simple and obvious that certain school knowledge belongs to the category of "unnecessary".But it is necessary to face, for example, the transport and the question of how to calculate the amount of cargo.You say that there is nothing easier?You are mistaken.Knowledge of calculation formulas, the concepts of "the density of matter", "bulk density bodies" are needed.
school knowledge - practical basis
Teachers of schools, teaching the basics of geometry, offer us a definition in volume of the space occupied by the body.This formula for determining the volume of recorded long ago, and you can find them in directories.Determine the volume of the body of regular shape mankind learned long before Archimedes' treatises.But this great Greek thinker has introduced a technique that gives the opportunity to determine the amount of any figure.His conclusions became the basis of integral calculus.Volumetric consider figures obtained during the rotation the flat geometric shapes.
Euclidean geometry with a certain accuracy to determine the amount of:
| geometric body | calculation formula | Basic parameters |
| a cuboid | V = lbh |
l - length, b - width, h - height |
| Cube | V = a3 | and - edge cube |
| cylinder | V = Sh | S - area of the base, h - height |
| Scope | V = 4πR3 / 3 | R- radius of the sphere |
difference between flat and three-dimensional figures can not be answeredSome sufferers question of how to calculate the amount of the rectangle.This is approximately the same as that found not know that.Confusion in the geometric material available, and the rectangle is sometimes called the cuboid.
being done, if the shape of the body is not so clearly defined?
Determination of complex geometric structures - work is not easy.It should be guided by several principles of immutable.
- Any body can be broken down into simpler parts.Displacement volume is the sum of its parts.
- Equivalent bodies have equal volumes, parallel transport of the bodies does not alter its scope.
- unit volume consider the volume of a cube with edge of length.
presence of bodies of irregular shape (remember the notorious crown king Heron) does not become a problem.Determining the volume of bodies hydrostatic weighing is possible.This is a process of directly measuring the volume of liquid to be shipped in its body, which will be discussed below.
various applications in the determination of the amount
Returning to the problem of how to calculate the volume of goods transported.What is the cargo: bulk or prepackaged?What are the parameters of packaging?There are more questions than answers.Another important issue will be the weight of the cargo carrying capacity is different as transport and the route - the maximum weight of the vehicle.Violation of rules of transportation threatens penalties.
Problem 1. Let a rectangular cargo containers filled with goods.Knowing the weight of the goods and the container, you can easily determine the total weight.The volume of the container determine the volume of a cuboid.
Knowing the capacity of transport, its dimensions, it is possible to calculate the possible volume of transported cargo.The correct balance of these parameters to avoid catastrophe, transport premature failure.
Task 2. Cargo - bulk material: sand, gravel and the like.At this stage, without the knowledge of physics can only do class specialist, whose experience in cargo transportation allows intuitively determine the maximum allowable amount for carriage.
scientific method presupposes a knowledge of such parameters as the density (bulk density) of cargo.
used formula V = m / ρ, where m - mass of the load, ρ - density of the material.Before you calculate the volume density of the cargo is found, which is also not difficult (tables, laboratory determination).
This technique also works fine when determining the volume of liquid cargo.At the same time as the unit of measurement used liter.
Determining the volume of construction forms
question of determining the volume plays an important role in the construction.The construction of houses and other structures - it expensive, building materials require careful and very precise calculation.
base of the building - the foundation - is usually cast construction, filled with concrete.Before you calculate the volume of concrete is necessary to determine the type of foundation.
slab foundation - a plate in the form of a cuboid.Columnar base - rectangular or cylindrical poles particular section.Determine the volume of post and multiplying it by the number, it is possible to calculate the cubic content of the concrete foundation of the whole.
calculating the amount of concrete for walls or ceilings, simply enter: define the scope of the entire wall, multiplying the length by the width and height, then separately define volumes of window and door openings.The difference between the volume of the total volume of the walls and openings - the volume of concrete.
How to determine the volume of the building?
Some applied problems require knowledge of the volume of buildings.These include the problem of repair, reconstruction, determining humidity, issues related to the heating and ventilation.
Before answering the question of how to calculate the volume of the building, make measurements on the outside of: cross-sectional area (length multiplied by width), height of the building from the bottom of the first floor to the attic.
determination of internal volume of heated space is carried by inland strokes.
device heating
modern apartments and offices is unthinkable without the heating system.The main part of the systems are battery and connecting pipes.How to calculate the volume of the heating system?The total volume of all sections of heating, which is indicated on the heat sink, you must add up the volume of the pipe.
At this stage there is a problem: how to calculate the volume of the tube.Imagine that pipe - cylinder solution comes by itself: using the formula for calculating the volume of a cylinder.In heating systems, pipes are filled with water, so it is necessary to know the area of the inner section of the tube.To do this, we define its inner radius (R).The formula determining the area of a circle: S = πR2.The total length of the pipes is determined by the length of the room.
Sewerage in the house - a system of pipes
Laying pipes for drainage, is also worth to know the volume of the tube.In this step, an external diameter similar to the foregoing steps.
Determination of metal, which is used to make pipes - as an interesting challenge.Geometrically pipe - cylinder cavities.Determine the area of the ring lying in her section - a task quite difficult, but solvable.A simpler way - to define the outer and inner volumes of pipes, the difference between these values and will be in volume of the metal.
Determination volumes in physics problems
famous legend of the crown of the king of Gerona has become known not only as a result of solving the problem of deducing "clean water" thieving jewelers.The result of complex mental activity Archimedes - quantification of the bodies of irregular geometry.The basic idea extracted philosopher - the amount of body fluid is displaced by the volume of the body.
In laboratory studies, use a graduated cylinder (beaker).Determine the volume of liquid (V1), it is immersed in the body, performing secondary measurement (V2).The volume is equal to the difference between the secondary and primary measurement: VT = V2 - V1.
This method of determining the volume of the bodies used in the calculation of bulk density of granular insoluble materials.It is extremely useful in determining the density of the alloy.
Calculate the amount of pins may be using this method.It seems rather difficult to define the scope of such a small body, like a pin or a pellet.Line it can not be measured, the measuring cylinder is also quite large.
But if you use exactly the same number of pins (n), it is possible by means of the graduated cylinder to determine their total volume (VT = V2 - V1).Then, the resulting value divided by the number of pins.V = VT \ n.
This problem becomes clear if one large piece of lead is necessary to cast a lot of pellets.
Units fluid volume
International System of Units involves measurement of the volume in m3.In everyday life increasingly used non-SI units: liter, milliliter.When it is determined how to calculate the volume in liters using the translation system 1 m3 = 1000 liters.
use in the daily life of other non-system actions can cause difficulties.The British used the more familiar for their barrels, gallons, bushels.
Drive System:
| British measures | Russian measures | ||
| bushel | 36,4 l | bucket | 12 l |
|
Gallon | 4,5 l | Barrel | 490 liters |
| Barrel (dry) | 115,628 liters | Shtof | 1, 23 L |
| barrels (oil) | 158, 988 liters | Charca | 0, 123 liters |
| English barrel for solids | 163,65 liters | Stiffy | 0,06 l |
Problems with non-standard data
Task 1. How to calculate the amount, knowing the height and area?Typically, this problem is solved by determining the amount of coverage of various parts galvanically.The area of the workpiece surface (S) is known.Layer thickness (h) - height.The volume is determined by product area and height: V = Sh.
Task 2. cubes interesting from a mathematical point of view, it seems the problem of determining the scope of, if you know the area of one face.It is known that the volume of the cube: V = a3, and where - the length of its faces.The area of the side surface of the cube S = a2.Extracting the square root of the square, we find the length of the face of the cube.Use the formula for the volume, we calculate its value.
Problem 3. Calculate the amount of pieces, if we know the area and are given some options.Additional parameters include the conditions of the aspect ratio, height, diameter at the base and more.
to solve specific problems need not only the knowledge of the formulas for calculating volumes, but other geometry formulas.
certain amount of memory
It is not related to the geometry of the task is to determine the amount of memory electronic devices.In today's computerized world is enough, this problem is not excessive.Exact devices What is a personal computer, do not tolerate approximate.
Languages amounts of memory stick or other storage device is useful when copying, moving information.
is important to know the amount of RAM and permanent memory.Often the user is faced with a situation where "is not a game," "hangs the program."The problem is quite possible with a low amount of memory.
unit of information is considered and its derivatives bytes (kilobytes, megabytes, terabytes).
1 KB = 1024 Bytes
1 MB = 1024 kB
1 GB = 1,024 MB
oddity in the system of allocation should be binary encoding information.
size memory storage device is its main characteristic.By comparing the amount of the transferred data and memory storage, one can determine its further exploitation.
term "volume" is so big, that to fully understand its versatility can only solving applied problems, interesting and exciting.
