text:
# Surge absorbers are used for protection against

1. High-voltage low-frequency oscillations
2. Low-voltage low-frequency oscillations
3. Low-voltage high-frequency oscillations
4. High-voltage high-frequency oscillations

Option 3 : Low-voltage high-frequency oscillations

## Protective Relays MCQ Question 1 Detailed Solution

Surge Absorbers:

• A surge absorber is a device to protect electrical equipment from over-voltage transients caused by external (lightning) or internal (switching) events.
• Surge absorber reduces the steepness of wave front of a surge.
• Surge absorbers are used for protection against low-voltage high-frequency oscillations.
• Surge arresters are not generally designed to protect against a direct lightning strike to a conductor, but rather against electrical transients resulting from lightning strikes occurring in the vicinity of the conductor.

# Which of the following should be used under no-load conditions?

1. Isolator
2. Rewireable fuse
3. Circuit breaker
4. Air-break switch

Option 1 : Isolator

## Protective Relays MCQ Question 2 Detailed Solution

Isolator:

• An isolator is a manually operated mechanical switch which separates a part of the electrical power.
• Isolators are used to open a circuit under no load.
• Its main purpose is to isolate one portion of the circuit from the other and is not intended to be opened while the current is flowing in the line.
• Isolators are generally used on both ends of the breaker so that repair or replacement of a circuit breaker can be done without any danger.

Rewireable fuse:

• A fuse is a part of the circuit that consists of a conductor that melts easily and breaks the connection when the current exceeds the predetermined value.
• The function of fuse wire is to carry the normal current without excessive heating but more than normal current when passes through fuse wire, it rapidly heats up and melts.
• The materials used for fuse wires are mainly tin, lead, zinc, silver, antimony, copper, aluminum, etc.
• Rewireable fuses are used to open a circuit under loaded conditions.

Circuit breaker:

• An electrical circuit breaker is a switching device which can be operated manually and automatically for controlling and protecting an electrical power system.
• During a short circuit fault or any other type of electrical fault, a high fault current will flow through this equipment as well as the power network itself.
• So a circuit breaker is needed to break the circuit and quenching the arc produced by the potential energy stored during braking.
• It is operated under loaded conditions.

Air-break switch:

• The switch whose contacts open in the air and quenching of an arc achieves by compressed air, such type of switch is called an air break switch.
• The air acts as a dielectric medium for the air-break switch.
• It is more effective and reliable as compared to another switch.
•  The maximum voltage for the switches is up to 35 kV.
• The air-break switches are classified into two types. They are Single-Pole Air-Break Switch and the Gang Operated Air-Break Switch.

# Which relay is also called angle impedance relay?

1. Impedance relay
2. Reactance relay
3. Mho relay
4. Frequency relay

Option 3 : Mho relay

## Protective Relays MCQ Question 3 Detailed Solution

Mho Relay:

A Mho relay measures a component of admittance |Y| ∠ θ. But its characteristic when plotted on the impedance diagram is a circle passing through the origin. It is also known as angle impedance relay.

The relay is called Mho relay because its characteristic is a straight line, when plotted on an admittance diagram.

Important Points:

• Impedance relay is a voltage restrained overcurrent relay.
• Reactance relay is an overcurrent relay with directional restraint.
• Mho relay is a voltage restrained directional relay.
• Reactance relay is suitable for the protection of a short transmission line because its operation is independent of arc resistance.
• The relay which is selected for a long transmission line should be less affected due to power swings. Hence Mho relay is preferred.
• Impedance relay is suitable for medium transmission lines.

# Which of the following relay is used for the protection of feeders and large busbars?

1. Under frequency relay
2. Buchholz relay
3. Distance relay
4. Differential relay

Option 4 : Differential relay

## Protective Relays MCQ Question 4 Detailed Solution

Differential relay:

• Differential relay operation depends on the phase difference of two or more electrical quantities.
• It works on the principle of comparison between the phase angle and the magnitude of the same electrical quantities.
• The differential relay is used for the protection of the feeder, large busbars, etc.

Distance relay:

• Distance relay is widely used for the protection of high-voltage AC transmission line and distribution lines.
• Distance protection schemes are commonly employed for providing the primary or main protection and backup protection for AC transmission line and distribution line against three-phase faults, phase-to-phase faults, and phase-to-ground faults.

Under frequency relay:

• Under frequency relay is used to protect the alternator when the frequency drops below the operating frequency.
• It is a backup protection for over fluxing (V/F) protection.
• Under frequency occurs due to turbine low speed, grid frequency fluctuation, etc.
• The generator can tolerate moderate under frequency operation provided voltage is within an acceptable limit.

Buchholz relay:

• Buchholz relay protects the transformer from internal faults.
• It is the gas actuated relay.
• The Buchholz relay is placed between the main tank and the conservator.
• Buchholz relay is used in the transformer having a rating higher than 500KVA.
• It is not used in a small transformer because of economic considerations.

# If for an IDMT relay with a plug setting of 50% and a CT ration of 400/5, the current is 3000 A, then the plug setting multiplier would be:

1. 7.5
2. 15.0
3. 18.75
4. 37.5

Option 2 : 15.0

## Protective Relays MCQ Question 5 Detailed Solution

Concept:

Plug setting multiplier:

Plug setting multiplier is the ratio of fault current and the product of plug setting and CT ratio.

Plug setting multiplier $$= \frac{{fault\;current}}{{plug\;setting \times CT\;ratio}}$$

Calculation:

Plug setting = relay setting × secondary CT current

= 0.5 × 5 = 2.5

CT ratio $$= \frac{{400}}{5} = 80$$

Plug setting multiplier $$= \frac{{3000}}{{2.5 \times 80}} = 15$$

# The relay used for feeder protection is:

1. Undervoltage relay
2. Translay relay
3. Thermal relay
4. Buchholz relay

Option 2 : Translay relay

## Protective Relays MCQ Question 6 Detailed Solution

Translay relay:

• The translay relay is a differential relay
• The arrangement is similar to overcurrent relay, but the secondary winding is not closed on itself
• These types of relays are used in the feeder protection and the scheme is called the Translay scheme
• In this scheme, two such relays are employed at the two ends of the feeder as shown in the Fig

• The secondaries of the two relays are connected to each other using pilot wires
• The connection is such that the voltages induced in the two secondaries oppose each other
• The copper coils are used to compensate the effect of pilot wire capacitance currents and unbalance between two currents transformers
• Under normal operating conditions, the current at the two ends of the feeder is the same
• The primaries of the two relays carry the same currents inducing the same voltage in the secondaries
• As these two voltages are in opposition, no current flows through the two secondaries circuits and no torque is exerted on the discs of both the relays
• When the fault occurs, the currents at the two ends of the feeder are different, hence unequal voltages are induced in the secondaries
• Hence the circulating current flows in the secondary circuit causing torque to be exerted on the disc of each relay
• But as the secondaries are in opposition, hence torque in one relay operates to close the trip circuit while in another relay the torque just holds the movement in an unoperated position
• The care is taken that at least one relay operates under the fault condition

# Which one of the given fuse is bigger_____?

1. DC
2. AC
3. DC or AC
4. AC and DC are same

Option 1 : DC

## Protective Relays MCQ Question 7 Detailed Solution

The fuse is the current interrupting devices which break or open the circuit by fusing the element and thus remove the faulty device from the main supply circuit.

Depending on the input supply voltage, fuses are the AC fuses and the DC fuses.

DC fuses:

• The DC fuse opens or breaks the circuit when the excessive current flow through it
• The only difficulty with the DC fuse is that the arc produced by the direct current is very difficult to extinct because there are no zero current flows in the circuit
• For reducing the DC fuse arcing, the electrodes are placed at more distance apart due to which the size of the fuse increases as compared to AC fuse

AC fuses:

The frequency of the AC fuses changes its amplitude from 0° to 60° in every one second. Thus, the arc extinction in the AC circuit can be done easily as compared to the DC circuit.

# A thyrite lightning arrester has

1. Inverse resistance characteristics
2. a gap
3. efficient earthing
4. a combination of inverse resistance characteristics and gap

Option 4 : a combination of inverse resistance characteristics and gap

## Protective Relays MCQ Question 8 Detailed Solution

• A thyrite lightning arrester has a combination of inverse resistance characteristics and gap
• Thyrite arrester is most common and is mostly used for the protection against high dangerous voltages
• The manufacturing processes have been perfected so that the electrical and mechanical characteristics can be duplicated or varied as desired within practical limits
• It does not follow Ohm's law, for each time the voltage is doubled the current increases 12.6 times
• Thus, it possesses nonlinear characteristics

# Which type of circuit breaker has a higher rating

1. miniature circuit breaker
2. Oil circuit breaker
3. Air blast circuit breaker
4. vacuum circuit breaker

Option 3 : Air blast circuit breaker

## Protective Relays MCQ Question 9 Detailed Solution

 Circuit breaker Rating Oil circuit breaker 3.3 kV – 220 kV Miniature circuit breaker 240 V Air blast circuit breaker 66 kV – 1100 kV Vacuum circuit breaker 3.3 kV – 33 kV

# The normal resistance of a pilot wire is:

1. More than 500 ohms
2. More than 1500 ohms
3. Less than 500 ohms
4. Less than 1500 ohms

Option 3 : Less than 500 ohms

## Protective Relays MCQ Question 10 Detailed Solution

Pilot wire:

• Whenever a transmission line or equipment is to be protected by using distance relay or differential relay or Merz-price protection. A wire is connected between the CT which is located at different ends of the protection zone.
• This wire provides the path for the circulating current produce in an abnormal condition, which is sensed by the relay and therefore is tripped; this wire is known as pilot wire.
• The normal resistance of a pilot wire is Less than 500 Ω.

# Match the items is List – I (Type of transmission line) with the items in List – II (Type of distance relay preferred) and select the correct answer using the codes given below the lists.    List – 1                                          a. Short Line                        b. Medium Line                                 c. Long Line                                        List - II1. Ohm Relay2. Reactance Relay     3. Mho Relay

1. a → 2, b → 1, c → 3
2. a → 3, b → 2, c → 1
3. a → 1, b → 2, c → 3
4. a → 1, b → 3, c → 2

Option 1 : a → 2, b → 1, c → 3

## Protective Relays MCQ Question 11 Detailed Solution

Distance Relay:

• A distance protection relay is a name given to the protection, whose action depends on the distance of the feeding point to the fault.
• The time of operation of such protection is a function of the ratio of voltage and current, i.e., impedance.
• This impedance between the relay and the fault depends on the electrical distance between them.
• Types of distance relays are impedance relays, reactance relays, and the mho relays.

 Transmission line Length &  Operating voltage Distance Relay Short-line (0-80) km (0-20) kV Reactance relay Medium line (80-200) km (20-100) kV Impedance relay Long-line (>200) km (>100) kV Mho relay

Important Points

• Reactance relay is suitable for the protection of a short transmission line because its operation is independent of arc resistance.
• The relay which is selected for a long transmission line should be less affected due to power swings. Hence Mho relay is preferred.
• Impedance relay is suitable for medium transmission lines.

# Which one of the following is NOT an example of Induction Type Relay?

1. Relaxation Pole Relays
2. Induction Cup Relays
4. Watt-Hour meter relays

Option 1 : Relaxation Pole Relays

## Protective Relays MCQ Question 12 Detailed Solution

Concept:

Induction type relay:

• The induction type relays are also called magnitude relays.
• These relays work on the principle of the induction motor or an energy meter.
• In these relays, a metallic disc is allowed to rotate between the two electromagnets.
• The coils of the electromagnets are energized with the help of alternating currents.
• The torque is produced in Induction relays due to the interaction of one alternating flux with eddy currents induced in the rotor by another alternating flux.
• The two fluxes have the same frequency but are displaced in time and space.
• As the interaction of alternating fluxes is the base of operation of Induction relays.
• These are not used for the d.c. quantities. These are widely used for protective relaying involving only a.c. quantities.

Based on the construction, the various types of induction relays are:

2)Watt-hour meter Relays
3)Induction cup Relays

Explanation:

So, The relaxation pole relay is not an example of an induction type relay.

# A distance relay with inherent directional property is known as

1. Buchholtz relay
3. Directional over current relay
4. Directional switched relay

## Protective Relays MCQ Question 13 Detailed Solution

A Mho relay or admittance relay measures a component of admittance |Y| ∠ θ. But its characteristic when plotted on the impedance diagram is a circle passing through the origin.

The relay is called Mho relay because its characteristic is a straight line, when plotted on an admittance diagram.

It is inherently a directional relay as it detects the fault only in the forward direction.

Important Points:

• Impedance relay is a voltage restrained overcurrent relay.
• Reactance relay is an overcurrent relay with directional restraint.
• Mho relay is a voltage restrained directional relay.
• Reactance relay is suitable for the protection of a short transmission line because its operation is independent of arc resistance.
• The relay which is selected for a long transmission line should be less affected due to power swings. Hence Mho relay is preferred.
• Impedance relay is suitable for medium transmission lines.

# There exist three terminals in a relay called C, NO, NC, in an application, it is desired that whenever the relay is active the load must be connected be supply, otherwise the load will be disconnected from supply. Which of the following connection type in relay will provide solution to the stated problem?

1. Supply to the NC terminal and load to the C terminal
2. Supply to the C terminal and load to the NO terminal
3. Supply to the C terminal and load to the NC terminal
4. Supply to the NC terminal and load to the NO terminal

Option 2 : Supply to the C terminal and load to the NO terminal

## Protective Relays MCQ Question 14 Detailed Solution

• Three terminals in a relay called C, NO, NC can be referred as
• C → Common
• NO → Normally open
• NC → Normally closed
• A condition is given in the question, that whenever the relay is active the load must be connected to the supply, otherwise disconnected from the supply.
• The above condition can be achieved only when the supply is connected to the ‘C’ terminal and load to the ‘NO’ terminal.

Note:

• If the load is connected to the ‘NC’ terminal of relay, then the load is always connected to the supply, whether the relay is active or not. However, it can be used to deactivate the load (by switching from ‘NC’ to ‘NO’). Which is the opposite case as given in the question.
• In this circuit diagram power wire switches from ‘NC’ to ‘NO” for activation of load.

# An ability of Relay system to operate under predetermined condition is called ________.

1. Sensitivity
2. Selectivity
3. Stability
4. Reliability

Option 1 : Sensitivity

## Protective Relays MCQ Question 15 Detailed Solution

Protective system:

A protection system is designed for the continuous operation of the power system and it should ensure maximum continuity of service with minimum damage to life, equipment, and property.

Objectives of system protection:

Speed of operation: Protection system should ensure minimum fault duration and consequently no equipment damage. If fault clearing is delayed then the system equipment will damage due to severe thermal and mechanical stress. so the protection system should detect the fault and clear the fault as quickly as possible.

Sensitivity: Protective relay should able to detect even the smallest possible value of fault current in the protection zone. 1.2 Ifl < Ipick up < Ifmin . It can be achieved with a proper setting to the protective system. Under this objective, the Relay system is designed to operate under a predetermined condition

Selectivity: The protection system should ensure max continuity of service with minimum system interruption. The protective relay should able to discriminate whether the fault is inside the zone or outside the zone of protection.

Reliability: It is the assurance that the protection system will operate 100% correctly. It has two aspects

1. Dependability: It should operate whenever it is required.
2. Security: It should not operate whenever it is not required.

Simplicity: The relaying system should have a minimum number of components and associated circuitry to achieve protection objectives.

Economical: A protection system should provide maximum protection at a minimum cost.

Relay: The relay is a switch that controls the circuits electromechanically. The main operation of this device is to make or break contact with the help of a signal without any human involvement. It is mainly used to control a high powered circuit using a low power signal.

# An effective method of providing protection to transmission lines against direct lightning strokes is by the use of

1. Earthing screen
3. Lightning arrester
4. None of these

Option 2 : Over head ground wires

## Protective Relays MCQ Question 16 Detailed Solution

• The most effective method of providing protection to transmission lines against direct lightning strokes are using overhead ground wires
• It provides considerable protection against direct lightning strokes on transmission lines
• A grounding wire provides a damping effect on any disturbance travelling along the line as it acts as a short-circuited secondary
• It provides a certain amount of electrostatic shielding against external fields
• Thus, it reduces the voltages induced in the line conductors due to the discharge of a neighbouring cloud

Important Point:

Earthing screen:

• The power stations and sub-stations generally have expensive equipment
• These stations can be protected against direct lightning strokes by providing an earthing screen
• It consists of a network of copper conductors (generally called shield or screen) mounted all over the electrical equipment in the substation or power station
• The shield is properly connected to earth on at least two points through a low impedance
• On the occurrence of a direct stroke on the station, the screen provides a low resistance path by which lightning surges are conducted to ground. In this way, station equipment is protected against damage
• The limitation of this method is that it does not provide protection against the travelling waves which may reach the equipment in the station

# Which of the following is a commonly used circuit breaker in India, for rural outdoor applications ranging from 22 kV to 66 kV?

1. Air Blast circuit breaker
2. Plain Break Oil Circuit Breaker
3. SF6 Circuit Breaker
4. Vacuum circuit breaker

Option 4 : Vacuum circuit breaker

## Protective Relays MCQ Question 17 Detailed Solution

For a country like India, where distances are quite large and accessibility to remote areas difficult. So, circuit breakers with easy installation and maintenance-free are more suitable in rural areas.

As vacuum circuit breakers are having this advantage, employed for outdoor applications ranging from 22 kV to 66 kV. Even with a limited rating of 60 to 100 MVA, they are suitable for most applications in rural areas.

Important Point:

• They are compact, reliable and have a longer life
• There are no fire hazards
• There is no generation of gas during and after the operation
• They can interrupt any fault current. The important feature of a VCB is that it can break any heavy fault current perfectly just before the contacts reach the definite open position.
• They require little maintenance and are quiet in operation
• They can withstand lightning surges
• They have low arc energy
• They have low inertia and hence require smaller power for the control mechanism

# A relay in a power system is a/an:

1. Automatic device that senses an abnormal condition in an electrical circuit and closes its contacts
2. Manual device that senses an abnormal condition in an electrical circuit and closes its contacts
3. Automatic device that senses an abnormal condition in an electrical circuit and opens its contacts
4. Automatic device that senses an abnormal condition in an electronic circuit and opens its contacts

Option 1 : Automatic device that senses an abnormal condition in an electrical circuit and closes its contacts

## Protective Relays MCQ Question 18 Detailed Solution

• relay is an automatic device which senses an abnormal condition of the electrical circuit and closes its contacts
• A relay which is operated on the principle of power flow in both sides of the line called directional relay
• Relays are commonly used to preventing excess power flow from a small generating station into a larger system.

Important Points

• In circuit breakers the device used for tripping the line in the event of short - the circuit is of the magnetic type
• At fault current levels, the magnetic field surrounding conductor provides sufficient force to unlatch the trip unit and trip the breaker

# The schematic arrangement given is of protection of alternator against:

1. earth fault
3. insulation failure of stator windings.
4. turn-to-turn fault in any single stator winding

## Protective Relays MCQ Question 19 Detailed Solution

The schematic arrangement given is of protection of alternator against unbalanced loading.

• Unbalanced loading means that there are different phase currents in the alternator.
• Unbalanced loading arises from faults to earth or faults between phases in the circuit external to the alternator.
• The unbalanced currents, if allowed to persist, may either severely burn the mechanical fixings of the rotor core or damage the field winding.

Construction and Operation:

• The scheme comprises three-line current transformers, one mounted in each phase, having their secondaries connected in parallel.
• A relay is connected in parallel across the transformer secondaries.
• Under normal operating conditions, equal currents flow through the different phases of the alternator and their algebraic sum is zero. Therefore, the sum of the currents flowing in the secondaries is also zero and no current flows through the operating coil of the relay.
• If unbalancing occurs, the currents induced in the secondaries will be different and the resultant of these currents will flow through the relay.
• The operation of the relay will trip the circuit breaker to disconnect the alternator from the system.

1. 0.1875 A
2. 0.2 A
3. 0.375 A
4. 60 kA


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# Power Series Odd Function Proof

## Homework Statement

Suppose that f(x)= summation an x^n for n = 0 to infinity for all x. If f is an odd function, show that a0 = a2 = a4 = ... = 0.

## The Attempt at a Solution

I said to consider sin(x), an odd function. When you do a series expansion only odd terms exist in the series so all even terms are equal to zero. This is because the nth derivative of sin(x) where n is an even number >= 2 always produces some form of cos(x) and when you plug 0 into cos(x) you get 0 and so even terms disappear. I was told that I restated the question and that I need to think harder about odd functions. I don't know how else to answer this. Thanks

SammyS
Staff Emeritus
Homework Helper
Gold Member
That's one EXAMPLE. It doesn't prove it's true for all such functions.

Suppose you have a function for which a0≠0, or a2≠0, or a4≠0, ...

Show that such a function is NOT odd.

What's true if a function is odd?

What's true if a function is NOT odd?

Dick
For odd functions, $f(-x) + f(x) = 0$. Using that will tell you $a_0 + a_2 x^2 + a_4 x^4 \cdots = 0$.

format: plain


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intsolve - Maple Help

intsolve

solve a linear integral equation

 Calling Sequence intsolve( Inteqn, funcn ) intsolve( Inteqn, funcn, method = ..., order = ... )

Parameters

 Inteqn - linear integral equation in funcn funcn - unknown function to be solved for method = ... - optional, the right-hand side can be any of: collocation, differentialequation, eigenfunction, Neumann, Laplace order = ... - optional, the right-hand side can be a positive integer, specifying the order of the series approximation to use with the Neumann method, or the degree of polynomial to use with the collocation method

Description

 • The intsolve command is able to find exact closed-form solutions to various types of integral equations as explained below. The solution is returned as an explicit equation with the unknown function in the left-hand side and the solution in the right-hand side. In some cases arbitrary constants are returned in the solution; they are represented as $\mathrm{_C1},\mathrm{_C2},\dots ,\mathrm{_Cn}$, as in the output of dsolve. When the solution contains integrals, they are represented with the inert Int.
 • By default, intsolve converts, using the differentialequation method, the linear integral equation into an equivalent ODE problem with initial values (ODE-IVP), then attempts solving this problem; if it fails, it uses an eigenfunction approach. The method Laplace is optional and tackles the problem using a Laplace transform. The collocation and Neumann methods are also optional, and compute approximations to the solution.
 • intsolve can recognize Fredholm and Volterra integral equations of the following kinds:
 Fredholm equation of the first kind (F1):

${\int }_{a}^{b}k\left(x,y\right)f\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆy=g\left(x\right)$

 Fredholm equation of the second kind (F2):

$f\left(x\right)+{\int }_{a}^{b}k\left(x,y\right)f\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆy=g\left(x\right)$

 Fredholm equation of the third (homogeneous) kind (F3):

$f\left(x\right)+{\int }_{a}^{b}k\left(x,y\right)f\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆy=0$

 Volterra equation of the first kind (V1):

${\int }_{a}^{x}k\left(x,y\right)f\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆy=g\left(x\right)$

 Volterra equation of the second kind (V2):

$f\left(x\right)+{\int }_{a}^{x}k\left(x,y\right)f\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆy=g\left(x\right)$

 Volterra equation of the third (homogeneous) kind (V3):

$f\left(x\right)+{\int }_{a}^{x}k\left(x,y\right)f\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆy=0$

 • The Laplace method is applicable to equations of V1, V2 and V3 type. One advantage to using this option is that the integral equation may contain the Dirac or Heaviside functions. This method is not suitable for finding a solution when the equation does not possess a convolution type kernel.
 • The collocation method for approximating solutions is applicable to Fredholm equations. This technique first partitions the interval $\left[a,b\right]$ uniformly into $n+1$ points, where positive integer $n$ is 10 by default or set by passing order=n. Then, the coefficients of an undetermined polynomial of degree $n$ are determined so that the polynomial satisfies the integral equation at the collocation points. The solve command is used to solve the collocation equations, so if a numeric approximation is desired, you can include floats inside the integral equation.
 • The Neumann method is applicable to equations of V2, F2, and some of V1 type. The default order of the Neumann series in the solution is Order - this value can be overridden by passing the optional argument order = n, where n is a positive integer.
 • When method is not specified, intsolve will check if the integral within the integral equation is of the form of a Fourier-Sine, Fourier-Cosine, or Laplace transform. That is, it will check if the integral equation can be written

${{\int }}_{0}^{\mathrm{\infty }}f\left(y\right)a\left(y\right)b\left(xy\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}y=c\left(x\right)$

 where $b\left(z\right)$ is one of $\mathrm{sin}\left(z\right)$, $\mathrm{cos}\left(z\right)$, or ${ⅇ}^{-z}$. In this case, intsolve will apply the appropriate inverse integral transform.
 • infolevel[intsolve] := 1,2,3 or 5 gives descriptive information on the progress of the computation. 5 may result in excessive printing.

Examples

Example 1

 • Consider the following example:
 > $\mathrm{eq1}≔p\left(x\right)-\frac{1}{2}\left(\mathrm{Int}\left(xyp\left(y\right),y=0..1\right)\right)=\frac{5}{6}x$
 ${\mathrm{eq1}}{≔}{p}{}\left({x}\right){-}\frac{\left({{\int }}_{{0}}^{{1}}{x}{}{y}{}{p}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right)}{{2}}{=}\frac{{5}{}{x}}{{6}}$ (1)
 > $\mathrm{intsolve}\left(\mathrm{eq1},p\left(x\right)\right)$
 ${p}{}\left({x}\right){=}{x}$ (2)
 If we instead use the 'Neumann' method, an approximation to this result is returned:
 > $\mathrm{intsolve}\left(\mathrm{eq1},p\left(x\right),\mathrm{method}=\mathrm{Neumann}\right)$
 ${p}{}\left({x}\right){=}\frac{{279935}{}{x}}{{279936}}$ (3)
 > $\mathrm{intsolve}\left(\mathrm{eq1},p\left(x\right),\mathrm{method}=\mathrm{Neumann},\mathrm{order}=20\right)$
 ${p}{}\left({x}\right){=}\frac{{21936950640377855}{}{x}}{{21936950640377856}}$ (4)

Example 2

 Computing the solution and verifying the result:
 > $\mathrm{eq2}≔f\left(x\right)=x+1+\mathrm{Int}\left(\left(1+2\left(x-y\right)\right)f\left(y\right),y=0..x\right)$
 ${\mathrm{eq2}}{≔}{f}{}\left({x}\right){=}{x}{+}{1}{+}{{\int }}_{{0}}^{{x}}\left({1}{+}{2}{}{x}{-}{2}{}{y}\right){}{f}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (5)
 > $\mathrm{intsolve}\left(\mathrm{eq2},f\left(x\right)\right)$
 ${f}{}\left({x}\right){=}{{ⅇ}}^{{2}{}{x}}$ (6)
 To verify this solution, transform it first into a procedure:
 > $f=\mathrm{unapply}\left(\mathrm{rhs}\left(\right),x\right)$
 ${f}{=}\left({x}{↦}{{ⅇ}}^{{2}{\cdot }{x}}\right)$ (7)
 > $\mathrm{eval}\left(\mathrm{eq2},\right)$
 ${{ⅇ}}^{{2}{}{x}}{=}{x}{+}{1}{+}{{\int }}_{{0}}^{{x}}\left({1}{+}{2}{}{x}{-}{2}{}{y}\right){}{{ⅇ}}^{{2}{}{y}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (8)
 > $\mathrm{value}\left(\right)$
 ${{ⅇ}}^{{2}{}{x}}{=}{{ⅇ}}^{{2}{}{x}}$ (9)

Example 3

 For some type of problems the solution obtained using the Laplace method contains Dirac functions:
 > $\mathrm{eq3}≔\mathrm{Int}\left(\mathrm{exp}\left(a\left(x-y\right)\right)f\left(y\right),y=0..x\right)=1$
 ${\mathrm{eq3}}{≔}{{\int }}_{{0}}^{{x}}{{ⅇ}}^{{a}{}\left({x}{-}{y}\right)}{}{f}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}{=}{1}$ (10)
 > $\mathrm{intsolve}\left(\mathrm{eq3},f\left(x\right),\mathrm{method}=\mathrm{Laplace}\right)$
 ${f}{}\left({x}\right){=}{\mathrm{Dirac}}{}\left({x}\right){-}{a}$ (11)

Example 4

 Another type of integral equation automatically solved using the eigenfunction approach:
 > $\mathrm{eq4}≔f\left(x\right)+b\left(\mathrm{Int}\left(\left(xy+{x}^{2}{y}^{2}\right)f\left(y\right),y=-1..1\right)\right)=d\left(x\right)$
 ${\mathrm{eq4}}{≔}{f}{}\left({x}\right){+}{b}{}\left({{\int }}_{{-1}}^{{1}}\left({{x}}^{{2}}{}{{y}}^{{2}}{+}{x}{}{y}\right){}{f}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right){=}{d}{}\left({x}\right)$ (12)
 > $\mathrm{intsolve}\left(\mathrm{eq4},f\left(x\right)\right)$
 ${f}{}\left({x}\right){=}{{\int }}_{{-1}}^{{1}}\left({-}\frac{{5}{}{{x}}^{{2}}{}{b}{}{{y}}^{{2}}}{{2}{}{b}{+}{5}}{-}\frac{{3}{}{x}{}{b}{}{y}}{{2}{}{b}{+}{3}}\right){}{d}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}{+}{d}{}\left({x}\right)$ (13)

Example 5

 An example where an arbitrary constant $\mathrm{_C1}$ is introduced with the solution:
 > $\mathrm{eq5}≔3{x}^{2}+4x=\mathrm{Int}\left(\left(6{x}^{2}y+4x{y}^{2}\right)p\left(y\right),y=-1..1\right)$
 ${\mathrm{eq5}}{≔}{3}{}{{x}}^{{2}}{+}{4}{}{x}{=}{{\int }}_{{-1}}^{{1}}\left({6}{}{{x}}^{{2}}{}{y}{+}{4}{}{x}{}{{y}}^{{2}}\right){}{p}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (14)
 > $\mathrm{intsolve}\left(\mathrm{eq5},p\left(x\right)\right)$
 ${p}{}\left({x}\right){=}{{x}}^{{2}}{}{\mathrm{_C1}}{+}\frac{{3}}{{4}}{}{x}{+}\frac{{3}}{{2}}{-}\frac{{3}}{{5}}{}{\mathrm{_C1}}$ (15)

Example 6

 An example which is in the form of an integral transform:
 > $\mathrm{eq6}≔\mathrm{Int}\left(\frac{f\left(y\right)}{1+{y}^{2}}\mathrm{sin}\left(xy\right),y=0..\mathrm{\infty }\right)=\mathrm{exp}\left(-3x\right)$
 ${\mathrm{eq6}}{≔}{{\int }}_{{0}}^{{\mathrm{\infty }}}\frac{{f}{}\left({y}\right){}{\mathrm{sin}}{}\left({x}{}{y}\right)}{{{y}}^{{2}}{+}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}{=}{{ⅇ}}^{{-}{3}{}{x}}$ (16)
 > $\mathrm{intsolve}\left(\mathrm{eq6},f\left(x\right)\right)$
 ${f}{}\left({x}\right){=}\frac{{2}{}{x}{}\left({{x}}^{{2}}{+}{1}\right)}{{\mathrm{\pi }}{}\left({{x}}^{{2}}{+}{9}\right)}$ (17)

Example 7

 The solution of a Fredholm integral equation can be approximated using the collocation method:
 > $a≔0:$
 > $b≔5:$
 > $g≔x↦-x:$
 > $k≔\left(x,y\right)↦\mathrm{piecewise}\left(x
 > $\mathrm{eq7}≔f\left(x\right)+\mathrm{int}\left(f\left(y\right)k\left(x,y\right),y=a..b\right)+g\left(x\right)$
 ${\mathrm{eq7}}{≔}{f}{}\left({x}\right){+}{{\int }}_{{0}}^{{5}}\left(\left\{\begin{array}{cc}{x}{}\left({5}{-}{y}\right)& {x}{<}{y}\\ {y}{}\left({5}{-}{x}\right)& {\mathrm{otherwise}}\end{array}\right\\right){}{f}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}{-}{x}$ (18)
 > $p≔\mathrm{unapply}\left(\mathrm{rhs}\left(\mathrm{intsolve}\left(\mathrm{eq7},f\left(x\right),\mathrm{method}=\mathrm{collocation},\mathrm{order}=10\right)\right),x\right)$
 ${p}{≔}{x}{↦}\frac{{220000}}{{8122882553}}{\cdot }{{x}}^{{10}}{-}\frac{{33081098947000}}{{60853654989179049}}{\cdot }{{x}}^{{9}}{+}\frac{{98331769350000}}{{20284551663059683}}{\cdot }{{x}}^{{8}}{-}\frac{{501031314879600}}{{20284551663059683}}{\cdot }{{x}}^{{7}}{+}\frac{{1594256521707500}}{{20284551663059683}}{\cdot }{{x}}^{{6}}{-}\frac{{6502492915408455}}{{40569103326119366}}{\cdot }{{x}}^{{5}}{+}\frac{{8429866033778125}}{{40569103326119366}}{\cdot }{{x}}^{{4}}{-}\frac{{9902325865977605}}{{60853654989179049}}{\cdot }{{x}}^{{3}}{+}\frac{{1401159656806875}}{{20284551663059683}}{\cdot }{{x}}^{{2}}{-}\frac{{231864099964442}}{{20284551663059683}}{\cdot }{x}$ (19)
 Plot of the approximate solution:
 > $\mathrm{plot}\left(p,a..b,\mathrm{title}="Approximate Solution of Integral Equation",\mathrm{color}=\mathrm{blue}\right)$
 To test how close the approximation matches the actual solution, we can sample the absolute value of the expression for the integral equation over the interval:
 > $n≔100:$
 > $q≔\mathrm{unapply}\left(\mathrm{abs}\left(\mathrm{eval}\left(\mathrm{eq7},f=p\right)\right),x\right):$
 > $X≔\mathrm{Vector}\left(n,i↦\mathrm{evalf}\left(a+\frac{\left(i-1\right)\cdot \left(b-a\right)}{n-1}\right)\right):$
 > $E≔\mathrm{~}\left[q\right]\left(X\right):$
 > $\mathrm{dataplot}\left(X,E,\mathrm{style}=\mathrm{line},\mathrm{title}="Error of Approximate Solution of Integral Equation"\right)$
 > $\mathrm{avg_error}≔\frac{\mathrm{add}\left(E\right)}{n}$
 ${\mathrm{avg_error}}{≔}{0.0001670862960}$ (20)
 > $\mathrm{max_error}≔\mathrm{max}\left(E\right)$
 ${\mathrm{max_error}}{≔}{0.001720013949}$ (21)

Example 8

 The Laplace method can be used to solve singular integral equations:
 > $\mathrm{eq8}≔\mathrm{Int}\left(\frac{f\left(y\right)}{\mathrm{sqrt}\left(x-y\right)},y=0..x\right)-{x}^{2}$
 ${\mathrm{eq8}}{≔}{{\int }}_{{0}}^{{x}}\frac{{f}{}\left({y}\right)}{\sqrt{{x}{-}{y}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}{-}{{x}}^{{2}}$ (22)
 > $\mathrm{intsolve}\left(\mathrm{eq8},f\left(x\right),\mathrm{method}=\mathrm{Laplace}\right)$
 ${f}{}\left({x}\right){=}\frac{{8}{}{{x}}^{{3}}{{2}}}}{{3}{}{\mathrm{\pi }}}$ (23)

References

 Chambers, LL. G. Integral Equations: A Short Course. London: International Textbook Company Ltd., 1976.
 Tricomi, F. G. Integral Equations. New York: Interscience, 1957.

Compatibility

 • The intsolve command was updated in Maple 2021.


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Documentation

# hjmvolspec

Specify Heath-Jarrow-Morton interest-rate volatility process

## Syntax

``VolSpec = hjmvolspec(Factor,Sigma_0)``
``VolSpec = hjmvolspec(Factor,CurveVol,CurveTerm)``
``VolSpec = hjmvolspec(Factor,Sigma_0,Lambda)``
``VolSpec = hjmvolspec(Factor,Sigma_0,CurveDecay,CurveTerm)``
``VolSpec = hjmvolspec(Factor,CurveProp,CurveTerm,MaxSpot)``

## Description

example

````VolSpec = hjmvolspec(Factor,Sigma_0)` creates a Constant volatility (Ho-Lee) structure for `hjmtree` by specifying the `Factor` as `'Constant'`. ```

example

````VolSpec = hjmvolspec(Factor,CurveVol,CurveTerm)` creates a Stationary volatility structure for `hjmtree` by specifying the `Factor` as `'Stationary'`.```

example

````VolSpec = hjmvolspec(Factor,Sigma_0,Lambda)` creates an Exponential volatility structure for `hjmtree` by specifying the `Factor` as `'Exponential'`.```

example

````VolSpec = hjmvolspec(Factor,Sigma_0,CurveDecay,CurveTerm)` creates a Vasicek, Hull-White volatility structure for `hjmtree` by specifying the `Factor` as `'Vasicek'`. ```

example

````VolSpec = hjmvolspec(Factor,CurveProp,CurveTerm,MaxSpot)` creates a Nearly proportional stationary volatility structure for `hjmtree` by specifying the `Factor` as `'Proportional'`.```

## Examples

collapse all

This example shows how to compute the `VolSpec` structure to specify the volatility model for `hjmtree` when volatility is single-factor proportional.

```CurveProp = [0.11765; 0.08825; 0.06865]; CurveTerm = [1; 2; 3]; VolSpec = hjmvolspec('Proportional', CurveProp, CurveTerm, 1e6)```
```VolSpec = struct with fields: FinObj: 'HJMVolSpec' FactorModels: {'Proportional'} FactorArgs: {{1x3 cell}} SigmaShift: 0 NumFactors: 1 NumBranch: 2 PBranch: [0.5000 0.5000] Fact2Branch: [-1 1] ```

This example shows how to compute the `VolSpec` structure to specify the volatility model for `hjmtree` when volatility is two-factor exponential and constant.

`VolSpec = hjmvolspec('Exponential', 0.1, 1, 'Constant', 0.2)`
```VolSpec = struct with fields: FinObj: 'HJMVolSpec' FactorModels: {'Exponential' 'Constant'} FactorArgs: {{1x2 cell} {1x1 cell}} SigmaShift: 0 NumFactors: 2 NumBranch: 3 PBranch: [0.2500 0.2500 0.5000] Fact2Branch: [2x3 double] ```

## Input Arguments

collapse all

Volatility factor, specified as a character vector with one of the following values:

• `'Constant'`

$\sigma \left(t,T\right)$ = `Sigma_0`

• `'Stationary'`

$\sigma \left(t,T\right)$ = `Vol(T- t)` = `Vol(Term)`

• `'Exponential'`

$\sigma \left(t,T\right)$ = `Sigma_0*exp(-Lambda*(T-t))`

• `'Vasicek'`

$\sigma \left(t,T\right)$ = `Sigma_0*exp(-Decay(T-t))`

• `'Proportional'`

$\sigma \left(t,T\right)$ = `Prop(T-t)*max(SpotRate(t),MaxSpot)`

### Note

You can specify more than one `Factor` by concatenating Factor names and their associated parameters.

Data Types: `char`

Base volatility over a unit, specified as a scalar numeric value.

Data Types: `double`

Decay factor, specified as a scalar numeric value.

Data Types: `double`

Number of curve `Vol` values at sample points, specified as a `NCURVES`-by`1` vector.

Data Types: `double`

Number of curve `Term` values at sample points, specified as a `NCURVES`-by-`1` vector.

Data Types: `double`

Number of curve `Decay` values at sample points, specified as a `NPOINTS`-by-`1` vector.

Data Types: `double`

Number of curve `Prop` values at sample points, specified as a `NCURVES`-by-`1` vector.

Data Types: `double`

Maximum spot rate, specified as a scalar numeric value.

Data Types: `double`

## Output Arguments

collapse all

Structure specifying the volatility model for `bktree`. `hjmvolspec` defines an HJM forward-rate volatility process based on the specified input `Factor`.

collapse all

### Volatility Process

The volatility process is $\sigma \left(t,T\right)$, where t is the observation time and T is the starting time of a forward rate.

In a stationary process, the volatility term is T–t. Multiple factors can be specified sequentially.

The time values T, t, and `Term` are in coupon interval units specified by the `Compounding` input of `hjmtimespec`. For instance if `Compounding` = `2`, `Term = 1` is a semiannual period (six months).


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# Sketch electric field of beamed antenna.

1. Sep 10, 2011

### Spinnor

Suppose we have an antenna which has large directional gain. Suppose the antenna broadcasts linearly polarized radiation in a narrow beam and at constant frequency. Sketch the electric field in the plane which contains both the electric field and the beam axis. Does the attached come close?

#### Attached Files:

• ###### electric field026.jpg
File size:
36.3 KB
Views:
78
Last edited: Sep 10, 2011
2. Sep 10, 2011

### Spinnor

Will the magnetic field look similar but rotated a quarter turn about the beam axis?

Given the electric field work backwards and sketch the vector potential.

Thanks for any help!

3. Sep 10, 2011

File size:
43.4 KB
Views:
57


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Dear friends, Please read our latest blog post for an important announcement about the website. ❤, The Socratic Team

# How do you convert 0.18 (18 repeating) to a fraction?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

40
Jun 5, 2018

$0.18181818 \ldots = \frac{2}{11}$

#### Explanation:

Let's begin by setting $x$ equal to our repeating fraction

$x = \setminus \textcolor{red}{0.18181818} \ldots$

The next step is a bit of a trick where we multiply by a power of 10 such that what remains to the right of the decimal is the repeating fraction itself, i.e.

$100 x = 18.1818181818 \ldots = 18 + \setminus \textcolor{red}{0.18181818} \ldots$

which can be re-written

$100 x = 18 + x$

subtracting $x$ from both sides we get

$99 x = 18$

and finally

$x = \frac{18}{99} = \frac{2}{11}$

• An hour ago
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• An hour ago


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# Loci in the complex plane

• Mar 15th 2009, 01:59 PM
Erghhh
Loci in the complex plane
Given that arg((z-1)/(z+1))=45deg.

find the centre and radius of the circle?

i equated z-1=z+1, but i'm pretty sure that was the wrong method as i didn't get the equation of a circle.

Starting point, anyone?
• Mar 15th 2009, 03:42 PM
Prove It
Quote:

Originally Posted by Erghhh
Given that arg((z-1)/(z+1))=45deg.

find the centre and radius of the circle?

i equated z-1=z+1, but i'm pretty sure that was the wrong method as i didn't get the equation of a circle.

Starting point, anyone?

$z = x + iy$

So $\frac{z - 1}{z + 1} = \frac{(x - 1) + iy}{(x + 1) + iy}$

Upon realising the denominator, this gives

$\frac{[(x - 1) + iy][( x+ 1) - iy]}{(x + 1)^2 + y^2}$

$= \frac{x^2 - 1 - ixy + iy + ixy + iy + y^2}{(x + 1)^2 + y^2}$

$= \frac{(x^2 + y^2 - 1) + 2iy}{(x + 1)^2 + y^2}$

$= \frac{x^2 + y^2 - 1}{(x + 1)^2 + y^2} + i\frac{2y}{(x + 1)^2 + y^2}$

Remember that $\arg{Z} = \arctan{\frac{\mathbf{I}(Z)}{\mathbf{R}(z)}}$.

Evaluate this and set it equal to $45^{\circ}$. This gives you the locus of the circle. You should then be able to find it's radius.
• Mar 15th 2009, 07:59 PM
mr fantastic
Quote:

Originally Posted by Erghhh
Given that arg((z-1)/(z+1))=45deg.

find the centre and radius of the circle?

i equated z-1=z+1, but i'm pretty sure that was the wrong method as i didn't get the equation of a circle.

Starting point, anyone?

At the moment I only have time to say the following:

This is not a circle. It's the major arc of a circle.

The arc joins z = 1 and z = -1 (these points are NOT included in the locus) and lies on a circle of radius $\sqrt{2}$ and centre located on the perpendicular bisector of the chord joining z = 1 and z = -1 at a distance of 1 from this chord.
• Mar 16th 2009, 02:43 AM
mr fantastic
Quote:

Originally Posted by Prove It
$z = x + iy$

So $\frac{z - 1}{z + 1} = \frac{(x - 1) + iy}{(x + 1) + iy}$

Upon realising the denominator, this gives

$\frac{[(x - 1) + iy][( x+ 1) - iy]}{(x + 1)^2 + y^2}$

$= \frac{x^2 - 1 - ixy + iy + ixy + iy + y^2}{(x + 1)^2 + y^2}$

$= \frac{(x^2 + y^2 - 1) + 2iy}{(x + 1)^2 + y^2}$

$= \frac{x^2 + y^2 - 1}{(x + 1)^2 + y^2} + i\frac{2y}{(x + 1)^2 + y^2}$

Remember that $\arg{Z} = \arctan{\frac{\mathbf{I}(Z)}{\mathbf{R}(z)}}$.

Evaluate this and set it equal to $45^{\circ}$. This gives you the locus of the circle. You should then be able to find it's radius.

This will give the circle upon which the locus lies. But the locus itelf is only a subset of this circle.

In fact, since $\frac{z - 1}{z + 1}$ cannot have an argument of $\frac{\pi}{4}$ unless its real and imaginary parts are BOTH positive, the solution has to satisfy the following two implied restrictions:

$Re \left[ \frac{z - 1}{z + 1} \right] > 0 \Rightarrow x^2 + y^2 > 1$ .... (1)

$Im \left[ \frac{z - 1}{z + 1} \right] > 0 \Rightarrow y > 0$ .... (2)

The locus is therefore the major arc of the circle I have previously described and which can be found by solving $Re \left[ \frac{z - 1}{z + 1} \right] = Im \left[ \frac{z - 1}{z + 1} \right]$. As well as this algebraic approach suggested by Prove It, the parameters of this circle can also be found from geometric considerations:

$arg(z - 1) - arg(z + 1) = \frac{\pi}{4}$ is the locus of the intersection point P of the rays $arg(z - 1) = \alpha$ and $arg(z + 1) = \beta$ under the restriction $\alpha - \beta = \frac{\pi}{4}$.

It's not hard to show that these two rays only intersect when $0 < \beta < \frac{3 \pi}{4}$, in which case the angle between these two rays is always $\frac{\pi}{4}$.

Using the the well known theorem that all angles subtended by the same chord of a circle at the circumference are equal, it follows that the locus is the major arc joining but not including z = -1 and z = 1. Using simple trigonometry on the right triangle AOC (where a is the point representing $z = -1$, O is the origin and C is the centre of the circle) it is easily seen that the arc lies on a circle with the following parameters:

1. Radius = $\frac{| -1 - 1|}{2 \sin \frac{\pi}{4}} = \sqrt{2}$.

2. Centre C located on the Imaginary axis at a distance $\frac{| -1 - 1|}{2 \tan \frac{\pi}{4}} = 1$ from the origin.

Several other approaches to finding the locus are possible, including a vector approach (using the dot product) and a parametric method.


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## Trigonometry (10th Edition)

$cot 1800^{\circ}= undefined$
1. What we know about 1800 degree angle $\theta=0^{\circ}$ (because it is a co-terminal angle and values of x and y will be the same) $x=1$ $y=0$ $r=1$ 2. Substitute in: $\cot\theta=\frac{x}{y}=\frac{1}{0}=undefined$


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# Reynolds Number Formula

## What is Reynolds Number?

Reynolds Number is the ratio of the internal forces to the viscous forces in a fluid. This ratio depends on several factors, such as the internal motion due to different fluid velocities. The Reynolds Number is important in mechanics and is a dimensionless quantity, i.e. it had no units. The Reynolds Number has several applications, we shall discuss the applications as we progress further. Its primary application is the prediction of change from laminar to turbulent flow.

### A Sneak Peek into The History

The concept of Reynolds number, internal forces, viscous forces was introduced by George Stokes back in the year 1851. However, there was no solid theory behind it. The idea was further polished and built upon in the year 1908 by Arnold Sommerfeld. The name Reynold comes from Osborne Reynolds, the man who popularized the use of Reynolds Numbers in the year 1883.

### The Formula for Reynolds Number

The formula for Reynolds Number is as given below

$Re = \frac{(ρuL)}{μ} = \frac{(uL)}{ν}$

In the formula given above,

ρ Is rho or the density of the given fluid.

u is the speed with which the fluid flows

μ is the dynamic viscosity of the fluid.

L is the characteristic linear dimension.

V is the kinematic velocity of the given fluid.

### The Significance of Reynolds Number

So now that you've learned the formula of the Reynolds Number and who exactly formulated the Reynolds Number and in what year, you must be wondering what does this mere number signify. Well, the Reynolds Number signifies the nature of the flow of a fluid across a cross-section. On a wide basis, there are two types of flowing styles that a liquid can flow in. One is laminar flow and the other is turbulent flow. Reynolds number is used to determine the type of flow of a fluid.

If Reynolds Number < 2300, then the liquid follows the laminar flow

If Reynolds number >2900, the liquid follows the turbulent flow

### Reynolds Number-Its Application in Fluid Mechanics

The Reynolds number is used to study fluids as they flow. The Reynolds number determines whether a fluid flow is steady or unsteady. (laminar and turbulent) If a flow is laminar, fluids will move along smooth streamlines. If the flow is turbulent, these streamlines break up and the fluid will move irregularly.

It is used when modelling the movement of organisms swimming through water. It is used to predict the transition from laminar to turbulent flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size version.

Reynolds number, in fluid mechanics, is a criterion of whether the fluid (liquid or gas) flow is steady (streamlined, or laminar) or on average steady with small unsteady fluctuations (turbulent).

The transition between fluids from laminar to turbulent occurs very suddenly.

Reynolds number is given by

 Reynolds Number = Inertial Force / Viscous Force

It is used when modeling the movement of organisms swimming through water. It is based on various factors like velocity, pressure, heat and temperature.

The Reynolds number is the ratio of inertial forces to viscous forces. The Reynolds number (Re) of a flowing fluid is computed by multiplying the fluid velocity by the pipe's internal diameter (to obtain the inertia force of the fluid) and then by dividing the result by the kinematic viscosity (viscous force per unit of  length).

### Number Range

Actually, the transition between laminar and turbulent flow occurs not at a specific value of the Reynolds number but in a range usually beginning between 1,000 to 2,000 and extending upward to between 3,000 and 5,000.

The maximum range is between 2300 to 4000.The Turbulent flow occurs over a range of Reynolds numbers from approximately 2,300 to 4,000, regardless of the nature of the fluid or the dimensions of the pipe or the average velocity. All that matters is that this specific combination of the parameters, known as the Reynolds number, fall in the range indicated.

### Types of Fluid Flow in Fluid Mechanics

The different types of fluid flow are:

2. Uniform and Non-Uniform Flow.

3. Laminar and Turbulent Flow.

4. Compressible and Incompressible Flow.

5. Rotational and Irrotational Flow.

6. One, Two and Three -dimensional Flow

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FAQs (Frequently Asked Questions)

1. In which areas the reynolds number formula is applicable?

The Reynolds number can be used to predict where this transition will take place. A vortex street around a cylinder. Reynolds number is applicable on fluids either be in liquid or gas form.For example if the candle is lit the Reynolds number will show the flame going from laminar to turbulent.It can also be applicable to wells, streams and lakes in liquid form. It is applicable to airplane,meteorological and climatic calculations.

2. On what factors does it depend?

R is the Reynolds number, which is unitless. It does depends on various factors like:

Flow Velocity:

If the  flow velocity increases, the Reynolds number also increases.

Factor Dimension:

If the dimension of a factor increases the characteristic dimension of the Reynolds number increases.

Fluid Density:

If the density of fluid decreases the value of Reynolds number also decreases.

It also depends on pressure, temperature, characteristic length and the mainstream velocity for gas.

3. In which chapter of physics do we learn about reynolds number?

In mechanical properties of physics in 11 std students learn about Reynolds formula, its application, its range, units and its dimensions. In this chapter they also learn how Reynolds number is applicable in liquids and gasses and their flow from steady to unsteady in fluids. They also learn the application of Reynolds number in various fields with examples.

4. What is fluid dynamics?

Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. Fluid dynamics has a great range of applications in various fields of studies like various streams of engineering, like civil, mechanical or chemical. It is also applicable to the study of oceans, aircraft's meteorology and chemical industries.

5. What is the function of the reynolds number?

Reynolds number is indicative of the relative importance of inertial and viscous effects in fluid motion. Explanation: Reynolds number constitutes an important criterion of motion in fluids depending on factors like heat transfer and temperature. It also considers the density of the fluid.

6. How is the reynolds number applicable in the pharmaceutical industry?

Purified water is the most used ingredient in pharmaceutical industries; it is used directly for the drug manufacturing process and also used for equipment's cleaning. If the Reynolds number is less than 2300 that means the flow of water is laminar.

7. What is the type of flow when the reynolds number lies between 2300 and 2900?

This is the confusing range of Reynolds numbers. It is the range where it is difficult to determine the type of flow. It is the range where the type of flow is transiting from laminar to turbulent. In this range of Reynolds number, the flow could either be laminar or turbulent. It isn't easy to determine. It depends on factors like the surface through which the fluid is flowing. The flow could at times be laminar and at times be turbulent. The flow in this range is dynamic and could switch types from time to time. Hence, it's called intermittent flow.

8. What does reynolds number depend on?

Reynolds number depends upon several factors. The factors on which the Reynolds Number depends are

• The density of the fluid: the density plays a key role in determining several physical factors of a fluid. Density is hence one of the primary factors that determine Reynolds Number.

• The speed with which the liquid is flowing in the cross-sectional area.

• Mu or the dynamic viscosity constant of the fluid.

• L, the length of the linear cross-section of the pipe.

9. What are the applications of reynolds number?

The Reynolds Number does have several practical applications in the real-time world! The Reynolds Number's primary application is the analysis of the flow of a fluid in a cross-sectional area. It helps determine the velocity at which the flow transits from laminar to turbulent. The Reynolds Number is applied in determining the behaviours of different liquids. For instance, the movement of glycerine is predicted in a circular duct. The Reynolds Number is observed to be below 2300, hence we can conclude that the flow is laminar!

Comment


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# Corrigendum

## Corrigendum to "Ion and electron heating during magnetic reconnection in weakly collisional plasmas" [J. Plasma Phys. 81, 305810201 (2015)]

There were errors in the evaluation of the spatial distribution of the energy dissipation described in our previously published paper [1]. The energy dissipation is induced by collisions as given by (2.5) in [1], where the collision term consists of the pitch angle scattering plus the energy diffusion and their gyro-diffusive corrections, and the conserving terms [2-3]. However, in [1], we only considered the contribution from the pitch angle scattering and its gyro-diffusive correction. Furthermore, the calculation of the gyro-diffusive correction included an error and was underestimated. We have improved the method to evaluate the spatial distribution of the energy dissipation as summarized in the separate note [4]. Here, we present the corrected version of Fig. 7, 9, 11, 13 in [1]. For all cases, patterns are same as those in [1] although the scales change slightly.

As described in [4], we still do not consider the contribution from the conserving terms since they are rather tedious. But, we have confirmed that the conserving terms have only minor effects to the energy dissipation in the nonlinear stage (see [4]). We also note that the time evoluions of the energy dissipation shown in Figs. 4-6 of [1] are calculated in a different way (which we call "the implicit evaluation"), so they do not change.

1. R. Numata and N. F. Loureiro, J. Plasma Phys. 81, 305010201 (2015).
2. I. G. Abel, M. Barnes, S. C. Cowely, W. Dorland., and A. A. Schekochihin, Phys. Plasmas 15, 122509 (2008).
3. M. Barnes, I. G. Abel, W. Dorland., D. R. Ernst, G. Hammett, P. Ricci, B. N. Rogers, A. A. Schekochihin and T. Tatsuno, Phys. Plasmas 16, 072107 (2009).
4. R. Numata, "Notes on AstroGK collision operators and heating diagnostics", unpublished note (2015). DOI: 10.13140/RG.2.1.5170.0563 [PDF]


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# Given a mixture of barium chloride, and potassium chloride, how would we separate one salt from the other?

Jul 8, 2016

By exploiting the relative insolubility of barium sulfate.

#### Explanation:

$B a S {O}_{4}$ is relatively insoluble in water; ${K}_{s p} = 1.08 \times {10}^{- 10}$.

So we mix stoichiometric barium chloride and potassium sulfate:

$B a C {l}_{2} \left(a q\right) + {K}_{2} S {O}_{4} \left(a q\right) \rightarrow B a S {O}_{4} \left(s\right) \downarrow + 2 K C l \left(a q\right)$

The supernatant solution should be $K C l \left(a q\right)$. Now this is filtered (or decanted) off to give a solution that should be reasonably pure $K C l \left(a q\right)$.

How to get the $K C l \left(a q\right)$ out of solution? Well you could suck it dry, but this would take forever. A better way would be to add ethyl alcohol to the solution. This solvent would precipitate out the $K C l$. This is collected on a frit, and subjected to extensive drying at elevated temperature.


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# Article

Full entry | PDF   (0.2 MB)
Keywords:
non-perfect group; \$G\$-conjugacy class; \$n\$-decomposable group
Summary:
Let \$G\$ be a finite group. A normal subgroup \$N\$ of \$G\$ is a union of several \$G\$-conjugacy classes, and it is called \$n\$-decomposable in \$G\$ if it is a union of \$n\$ distinct \$G\$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.
References:
[1] Ashrafi, A. R.: On decomposability of finite groups. J. Korean Math. Soc. 41 (2004), 479-487. DOI 10.4134/JKMS.2004.41.3.479 | MR 2050157 | Zbl 1058.20026
[2] Ashrafi, A. R., Sahraei, H.: On finite groups whose every normal subgroup is a union of the same number of conjugacy classes. Vietnam J. Math. 30 (2002), 289-294. MR 1933567 | Zbl 1018.20026
[3] Ashrafi, A. R., Venkataraman, G.: On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes. Proc. Indian Acad. Sci., Math. Sci. 114 (2004), 217-224. DOI 10.1007/BF02830000 | MR 2083462 | Zbl 1070.20027
[4] Gorenstein, D.: Finite Groups. Chelsea Publishing Company, New York (1980). MR 0569209 | Zbl 0463.20012
[5] Guo, X., Chen, R.: On finite \$X\$-decomposable groups for \$X=\{1, 2, 3, 4\}\$. Bull. Iranian Math. Soc. 40 (2014), 1243-1262. MR 3273835 | Zbl 06572891
[6] Guo, X. Y., Li, J., Shum, K. P.: On finite \$X\$-decomposable groups for \$X=\{1, 2, 4\}\$. Sib. Math. J. 53 (2012), 444-449 translation from\global\questionmarktrue Sib. Mat. Zh. 53 558-565 2012. DOI 10.1134/S0037446612020255 | MR 2978574 | Zbl 1257.20031
[7] Isaacs, I. M.: Character Theory of Finite Groups. Dover Publications, New York (1994). MR 1280461 | Zbl 0849.20004
[8] Riese, U., Shahabi, M. A.: Subgroups which are the union of four conjugacy classes. Commun. Algebra 29 (2001), 695-701. DOI 10.1081/AGB-100001534 | MR 1841992 | Zbl 0990.20020
[9] Rose, H. E.: A Course on Finite Groups. Universitext, Springer, London (2009). DOI 10.1007/978-1-84882-889-6 | MR 2583713 | Zbl 1200.20001
[10] Shi, W. J.: A class of special minimal normal subgroups. J. Southwest Teachers College 9 (1984), 9-13 Chinese.
[11] Wang, J.: A special class of normal subgroups. J. Chengdu Univ. Sci. Technol. 1987 (1987), 115-119 Chinese. English summary. MR 1028900 | Zbl 0671.20022

Partner of


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# What is faster: many ifs, or else if in PHP?

An else if is a better option.

Below is a sample code for multiple if statements −

if(condition_A){
//perform some action
}
if(condition_B){
//perform some action
}

Below is a sample code for else if statement −

if(condition_A){
//perform some action
}
else if(condition_B){
//perform some action
}

When else if statements are used, if a condition is satisfied, the checking stops there and the operations associated with the condition are executed. This way, the operations and conditions get over quickly.


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Race Your Marbles to Discover a Liquid's Viscosity

The viscosity o a fluid is a meisur o its reseestance tae gradual deformation bi shear stress or tensile stress. References This page wis last eeditit on 11 Mey 2017, at 02:22. viscosity | Definition, Facts, & Examples | Britannica Viscosity, resistance of a fluid (liquid or gas) to a change in shape, or movement of neighbouring portions relative to one another. Viscosity denotes opposition to flow. The reciprocal of the viscosity is called the fluidity, a measure of the ease of flow. Molasses, for example, has a greater viscosity than water. Intrinsic viscosity - Wikipedia Intrinsic viscosity [] is a measure of a solute's contribution to the viscosity of a solution.It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to the mass concentration of the polymer.. Intrinsic viscosity is defined as [] = → −where is the viscosity in the absence of the solute, is (dynamic or kinematic) viscosity of

## Product, Liquid or Substance: Viscosity in CPS (mPas) CHEMICALS: Acetone: 0.6: Acetone Slurry: 60000: Acrylic and Water: 2500: Alkali Cellulose Slurry: 60: Anhydrous

The viscosity of a fluid is the measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the informal concept of "thickness": for example, honey has a higher viscosity than water. Viscosity is the property of a fluid which opposes the relative motion between two surfaces of the fluid that are moving at different velocities. Viscosity - Chemistry LibreTexts The viscosity is calculated with Equation $$\ref{1}$$ $\eta =K t \label{1}$ where $$K$$ is the value of a liquid with known viscosity and density such as water. Once the value of K is known, the viscosity can be determined by measuring the amount of time the test liquid flows between the two graduated marks. Units of Measure: viscosity - Wiktionary

### Nov 26, 2016

Saybolt universal viscosity (SUV), and the related Saybolt FUROL viscosity (SFV), are specific standardised tests producing measures of kinematic viscosity. FUROL is an acronym for fuel and road oil. Saybolt universal viscosity is specified by the ASTM D2161. Both tests are considered obsolete to other measures of kinematic viscosity, but their results are quoted widely in technical literature. Glycerol - Wikipedia Glycerol (or glycerine, glycerin) is a simple polyol (succar alcohol) compoond.It is a colorless, odorless, viscous liquid that is widely uised in pharmaceutical formulations.Glycerol haes three hydroxyl groups that are responsible for its solubility in watter an its hygroscopic naitur. The glycerol backbone is central tae aw lipids kent as triglycerides. Kinematic Viscosity Formula - Softschools.com Kinematic viscosity is the measure of the inherent resistance of a fluid to flow when no external force is exerted, except gravity. It is the ratio of the dynamic viscosity to its density, a force independent quantity. Kinematic viscosity can be obtained by dividing the absolute viscosity of … What Is Viscosity? - Precision Cleaners Chemists have a slightly different feel for the phenomenon; viscosity is defined as its resistance to gradual deformation. This corresponds to the informal concept of “thickness” — for example, honey is thicker and has a much higher viscosity than water. According to Wikipedia, viscosity is the friction between the molecules of fluid.


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We will be translating the sentence "twice y differs from 5 by more than two, “into a mathematical equation. The function appears to cross the x axis at x = 2, x = -1, and x = -4. You can test out of the Get access risk-free for 30 days, If we "assign" to $x$ the "value" $d_1$ and to $y$ the "value" $d_2$ and finally to $z$ the "value" $d_3$, we have that: Thus, we have a conditional: $\text{true } \to \text { false}$ that is $\text {false}$, for the said values. Not sure what college you want to attend yet? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. How is “interpretation” used differently in propositional vs. first-order logic? Create an account to start this course today. Open Sentences: A triangle has n sides. The number e can also be defined as the limit as n goes to infinity of (1+ 1/n)^n. There's an exercise in these slides with answers, but I'm having trouble understanding what makes them correct. just create an account. What type of function is it? Each break in the flow of the line tells you that the formula has changed at that point. Sentence case is the conventional way of using capital letters in a sentence or capitalizing only the first word and any proper nouns. Examples of different interpretation in a sentence, how to use it. It only takes a minute to sign up. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Making statements based on opinion; back them up with references or personal experience. Node version error during Salesforce DX pre-release plugin installation. Mathematical expressions containing the "equal" sign are considered equations; those containing the "inequality" symbol are … are all useful pieces of information toward understanding the function. Where does the function cross the x and y axes? When you're interpreting a function, you're answering questions based on the occasionally cryptic information available. There are three different sections (shown in different colors in the picture to help you visualize them), which behave in different ways. Philosophy of mathematics, branch of philosophy that is concerned with two major questions: one concerning the meanings of ordinary mathematical sentences and the other concerning the issue of whether abstract objects exist. Using the previous problem, complete the sentence below for the interpretation of the 95% confidence interval. A mathematical function consists of three parts: input, output and the relationship between the two, which can be depicted on a graph. Like letters in the alphabet, they can be used to form words, phrases and sentences that would constitute a larger part of the mathematical lexicon. So here's my question: What is it in this interpretation that makes sentence a false, and the others true? Functions on graphs often take on a specific shape, such as a parabola, or curved line. Usually, once you get the math equation, you're fine; the actual math involved is often fairly simple. I[a] = d3 In the 3rd degree polynomial graph above, we can see that the function appears to have a local maximum at x = -2.8 and a local minimum at x = 0.8. What is the daytime visibility from within a cloud? (a) ∀x∀y∀z((P(x, y) ∧ P(y, z)) → P(x, z)) Interpret definition, to give or provide the meaning of; explain; explicate; elucidate: to interpret the hidden meaning of a parable. Polynomials are generated by addition, multiplication, and raising to the power of a positive integer. is a sentence. 102 lessons However, every number sentence gives us information, and based on the information provided; it is possible to change the statement from false to true. Plotting important points, such as the intercepts and turning points, is also part of interpreting a function. While 5 x 2 = 10 produces an accurate sentence, 2 x 5 = 01, or 2 5 x = 10, or 2 = 5 x 10 demonstrate errors in coding, syntax, and information, respectively. Transcendental functions are not algebraic. In general, a mathematical statement consists of two parts: the hypothesis or assumptions, and the conclusion. An interpretation of something is an opinion about what it means. Select a subject to preview related courses: If we look back at the graph, we realize that it doesn't cross the y axis at -8 (which is where the above equation would if all the xs were 0), so we have to modify our equation to make it fit the line shown. mathematics definition: 1. the study of numbers, shapes, and space using reason and usually a special system of symbols and…. The end appears to extend toward negative infinity (downward) on the left and positive infinity (upward) on the right of the graph. first two years of college and save thousands off your degree. Title: Meaning of division worksheet Author: K5 Learning Subject: Grade 3 Division Worksheet Keywords: numbers, math, worksheet, grade 3, exercise, division Introduction to confidence intervals Interpreting confidence levels and confidence intervals AP.STATS: For example: 3 + 5 = 8 = 1 + x + x^2/2 + x^3/3! Professor Wolfgang gives the data a very different interpretation. Let’s write an algebraic expression, described with a sentence: The sum of twice a number and six As you know the first operation in the sentence (the sum), is the last one to be performed. Example: 15 is divisible by 3, because 15 ÷ 3 = 5 exactly But 9 is not divisible by 2 because 9 ÷ 2 is 4 with 1 left over. Interpreting a mathematical function means obtaining meaningful information from a line graph or given symbolic representation. It differs from the Reading Comprehension test in that it requires you to identify the meaning of simple phrases or collections of two sentences. I'm currently learning about interpretations in first-order logic and I found some slides online about the subject which were very helpful. It's a piece function. Synonyms: explanation , meaning , reading , understanding More Synonyms of interpretation Similar to the Reading Comprehension test, the ESL Sentence Meaning section of the exam tests your ability to comprehend sentence meaning and intent. You have to stick to the order of operations. This sentence is true in the positive real numbers ℝ +, false in the real numbers ℝ, and true in the complex numbers ℂ. (In plain English, this sentence is interpreted to mean that every member of the structure concerned is the square of a member of that … Mathematics is a way to model the world around us, from these very basic addition and subtraction models to advanced Calculus, so understanding that each number and symbol in the “math sentence” represents a real thing and has meaning is an essential skill! site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. A mathematical equation may be stated in words to form a sentence that has a noun and a verb, just like a sentence in a spoken language. The hardest thing about doing word problems is using the part where you need to take the English words and translate them into mathematics. She has done research and teaching in mathematics and physical sciences. How should I handle the problem of people entering others' e-mail addresses without annoying them with "verification" e-mails? All rights reserved. 10 chapters | How to evaluate truth of expressions with free variables? Dr. Chan has a Ph.D. from the U. of California, Berkeley. Clearly, the intended meaning of each mathematical sentence is governed by the same components that affect the accuracy of English sentences. Therefore, a number sentence does not necessarily have to be true. and career path that can help you find the school that's right for you. Computation definition is - the act or action of computing : calculation. Log in here for access. To learn more, see our tips on writing great answers. Get the unbiased info you need to find the right school. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. | {{course.flashcardSetCount}} Synonym Discussion of interpret. Developing student’s language skills by focusing on mathematical vocabulary and using sentence frames helps students learn the language of math! Correlation vs. Causation: Differences & Definition, Quiz & Worksheet - Interpreting Mathematical Functions, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, Functions: Identification, Notation & Practice Problems, How to Add, Subtract, Multiply and Divide Functions, Steps for Building Functions in Mathematics, Scatterplots and Line Graphs: Definitions and Uses, GACE Program Admission Assessment Test II Mathematics (211): Practice & Study Guide, Biological and Biomedical Spelling measures an individual's ability to encode sounds into written form through the use of a dictated spelling format containing both letters and words. When x is 0, the function value will be -4, so that's the y intercept. Thanks for contributing an answer to Mathematics Stack Exchange! Identify location of old paintings - WWII soldier. Already registered? 1. Plugging in the data point they gave me, and solving for the value of k, I get: Now that I have the value of k, I can plug in the new values, and solve for the new value of y: Then the answer is: Affiliate. Sentence frames give students an opportunity to access the math concepts and to engage in the classroom conversation of math by answering questions. What does the term "svirfnebli" mean, and how is it different to "svirfneblin"? 3. Example of a first-order interpretation Domain: A chess set Individual constants: a: The white King b: The black Queen c: The white King's pawn F (x): x is a piece G (x): x is a pawn H (x): x is black I (x): x is white J (x, y): x can capture y Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education. (b) ∀x∀y((P(x, y) ∧ P(y, x)) → x = y) It appears to cross the y axis at y = -2. A sentence … What is the fallacy in my reasoning of this sentence-to-first-order-logic-problem? T he language and vocabulary of mathematics contain a large amount of symbols — some being more technical than others. Meaning of Multiplication Sentences Author: K5 Learning Subject: Grade 3 Multiplication Worksheet - meaning of multiplication Keywords: Multiplication sentences, grade 3, math, meaning Created Date: 7/22/2020 10:12:06 AM Why is the sentence $∀x∀y∀z((P(x, y) ∧ P(y, z)) → P(x, z))$ [call it : (A)] false in the said interpretation ? © copyright 2003-2021 Study.com. The Elementary functions are algebraic or non-algebraic (Transcendental). flashcard set{{course.flashcardSetCoun > 1 ? A mathematical sentence can also use symbols or words like equals, greater than, or less than. Learn more. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The word twice generally means two times, so 2*y (because y comes next in the sentence). Most mathematical statements you will see in first year courses have the form "If A, then B" or "A implies B" or "A $\Rightarrow$ B". First order logic - is this a valid interpretation of a sentence? The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a " meaning " or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old algebraic problem became inevitable, and the true solution, now so obvious, was eventually obtained. Interpreting a function means converting the symbols of a formula or a drawn graph into meaningful information that fits what you're looking for. +...... k! Translating the formula from English to math, I get: y = kxz. Log in or sign up to add this lesson to a Custom Course. n is the degree of the root. Because in the said interpretation, with domain $D$, we have that the (binary) predicate letter $P$ is interpreted with the relation : Consider now the formula : $(P(x, y) ∧ P(y, z)) → P(x, z)$ [call it : (B)]. In most newspapers in the U.S. and in virtually all publications in the U.K., sentence case, also known as down style and reference style , … The domain and range may be infinite, since the graph probably continues beyond what is shown. I'm not seeing 'tightly coupled code' as one of the drawbacks of a monolithic application architecture, Maximum useful resolution for scanning 35mm film. Possible comparison operators are "equal," "not equal," "greater than," "less than," "greater than or equal to" and "less than or equal to." This means the graph will be some version of the polynomial you get when you multiply three binomials together: (x - 2)(x + 1)(x + 4) = x³ + 3x² - 6x - 8. 's' : ''}}. Can be true or false (depending on the value of z) 3y = 4x + 2. This means the graph will look like a parabola, a curved line that has a simple curve in the middle and goes in the same direction at both ends. | 9 Can be true or false (depending on the values of x and y) a + b = c + d. Can be true or false (depending on the values of a,b,c,d) A root of degree 2 is called a square root. To unlock this lesson you must be a Study.com Member. What is the highest road in the world that is accessible by conventional vehicles? Earth and moon gravitational ratios and proportionalities. No. In this lesson, we will explore how to obtain meaningful information from graphs and function formulas. 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The first is a straightforward question of interpretation: What is the However, there could also be number sentences which say: 12 + 6 = 9 is not true, but 12 + 6 = 18 is true. Use MathJax to format equations. Giving students part of the language of a sentence, or the structure of a sentence, allows a focus on the concept, not the language. Once you know the shape and the intercepts, plotting a few more points by putting in values for x will give you an acceptable graph: Interpreting functions from graphs and definitions require that you convert the picture or symbols that you have into the information that you need. Services. To learn more, visit our Earning Credit Page. Can be true or false (depending on the value of n) z is a positive number. Differs normally means is different and a difference from something in a mathematical sentence means subtraction most of the time, thus minus five. 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Algebraic functions (polynomials, rational functions, and nth roots), use addition, multiplication and the usual algebraic operators to define themselves. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. credit by exam that is accepted by over 1,500 colleges and universities. flashcard sets, {{courseNav.course.topics.length}} chapters | MathJax reference. Rational functions are the ratio of two polynomials. Are the longest German and Turkish words really single words? I[b] = d2. Interpret definition is - to explain or tell the meaning of : present in understandable terms. 2. Sciences, Culinary Arts and Personal Having found in the interpretation some values for $x,y,z$ that "falsify" the formula (B), we have shown that it is not true for all possible values, and thus the formula (A) is false in that interpretation. I've already read a lot about interpretations online, and even understood simpler examples, but this one seem to be a bit more advanced than those. "Get used to cold weather" or "get used to the cold weather"? All polynomials cross the x axis when the function value goes to 0. The 7th grade and 8th grade exercise has eight sentences with three-part phrasal verbs. Since the highest exponent of the function is a 2, it's a quadratic function. study Enrolling in a course lets you earn progress by passing quizzes and exams. For convenience, the slides can be found at: http://iaoa.org/isc2014/uploads/SlidesWassermann02.pdf, For each of the following sentences, give a logical interpretation Study.com has thousands of articles about every When dividing by a certain number gives a whole number answer. Can Pluto be seen with the naked eye from Neptune when Pluto and Neptune are closest? In short: Why isn't sentence (a) true and why aren't sentence (b) and (c) false in this interpretation? Since it crosses at -8 instead of the -2 shown in the graph, our equation is similar to the one shown, but is too steep. Convert Phrasal Verbs into Normal Verbs Help your children stretch their phrasal verb horizons, by learning how to convert the meaning of a phrasal verb into a normal verb. rev 2021.1.18.38333, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Understanding interpretation First-Order Logic, http://iaoa.org/isc2014/uploads/SlidesWassermann02.pdf. Visit the GACE Program Admission Assessment Test II Mathematics (211): Practice & Study Guide page to learn more. 18 examples: He concludes that the contradiction stems from a different interpretation of… How can a GM subtly guide characters into making campaign-specific character choices? How to use computation in a sentence. In this equation, that would be true at x = 2 and x = -2, so those would be the x intercepts. Maximum points, minimum points, domain, range, x intercepts, y intercepts, and end behavior (which directions do the ends of the graph go?) Instead of creating a wall of text, I'm just going to post the interpretation which makes sentence (a) false and sentences (b) and (c) true: D = {d1, d2, d3} A mathematical sentence combines two expressions with a comparison operator to create a fact that may be either true or false. We know that at those values for x, the function output goes to 0. exp(x) = sum from k=0 to infinity of x^k/k! How to use interpret in a sentence. See more. Prove that the following is true: a sentence is unsatisfiable if and only if it implies all other sentences. There are power functions, exponential functions, logarithms, hyperbolic functions and periodic functions. I am so focused on developing my students’ math skills with the Standards for Mathematical Content that I forget that I also need to implement (and assess) the Mathematical Practice Standards . - Definition, Causes, Properties & Characteristics, Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers, Working Scholars® Bringing Tuition-Free College to the Community. Here we are using the = sign which indicates a balance of both sides. 10 + 5 = 15. Asking for help, clarification, or responding to other answers. In addition, each of the three sections is a straight line, which means you're not dealing with any exponents larger than 1. But figuring out the actual equation can seem nearly impossible. Court Interpreting Classes and Courses Overview, Court Interpreter Training and Education Program Info. The line climbs the left side of the graph (. imaginable degree, area of Is direction of logical consequence in FOL arbitrary? credit-by-exam regardless of age or education level. By using sentence frames in math, teachers can offer a method of scaffolding for students as they build and develop math skills. Why are good absorbers also good emitters? Sentence Comprehension measures an individual's ability to gain meaning from words and to comprehend ideas and information contained in sentences through the use of a modified cloze technique. I[P] = {(d1, d2),(d2, d3)} The basic exponentiation identity is exp(x+y) = exp(x) exp(y). Did you know… We have over 220 college It only takes a minute to sign up. Create your account. In the 3rd degree graph shown, the places where it crosses the x axis are keys to the formula. 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I am 95% confident that_____ to _____ percent of people strongly agree that K-12 schools should teach critical thinking to school children. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons All other trademarks and copyrights are the property of their respective owners. What follows is a list of hints and helps. Question 1 ESL Sentence Meaning Practice Test for the ACCUPLACER ® Test Choose the correct word for the blank in this sentence: Michael looked for oncoming traffic then walked ____ the street to … For example, if you are told that f(x) = x² - 4, you could answer the following questions to interpret it: 1. For example, perhaps we want to answer the following questions about the three-region graph pictured above: Looking at the graph, we can figure out that: The function type will tell you what kind of formula or relationship you're working with and the general behavior you can expect from the function. 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blob: f3c687dedbdb72912d33e35082217c1e87c5551b (plain)
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103  /********************************************************************** version.c - $Author$ created at: Thu Sep 30 20:08:01 JST 1993 Copyright (C) 1993-2007 Yukihiro Matsumoto **********************************************************************/ #include "ruby/ruby.h" #include "version.h" #include #ifndef EXIT_SUCCESS #define EXIT_SUCCESS 0 #endif #define PRINT(type) puts(ruby_##type) #define MKSTR(type) rb_obj_freeze(rb_usascii_str_new(ruby_##type, sizeof(ruby_##type)-1)) const int ruby_api_version[] = { RUBY_API_VERSION_MAJOR, RUBY_API_VERSION_MINOR, RUBY_API_VERSION_TEENY, }; const char ruby_version[] = RUBY_VERSION; const char ruby_release_date[] = RUBY_RELEASE_DATE; const char ruby_platform[] = RUBY_PLATFORM; const int ruby_patchlevel = RUBY_PATCHLEVEL; const char ruby_description[] = RUBY_DESCRIPTION; const char ruby_copyright[] = RUBY_COPYRIGHT; const char ruby_engine[] = "ruby"; VALUE ruby_engine_name = Qnil; /*! Defines platform-depended Ruby-level constants */ void Init_version(void) { VALUE v = MKSTR(version); /* * The running version of ruby */ rb_define_global_const("RUBY_VERSION", v); /* * The date this ruby was released */ rb_define_global_const("RUBY_RELEASE_DATE", MKSTR(release_date)); /* * The platform for this ruby */ rb_define_global_const("RUBY_PLATFORM", MKSTR(platform)); /* * The patchlevel for this ruby. If this is a development build of ruby * the patchlevel will be -1 */ rb_define_global_const("RUBY_PATCHLEVEL", INT2FIX(RUBY_PATCHLEVEL)); /* * The SVN revision for this ruby. */ rb_define_global_const("RUBY_REVISION", INT2FIX(RUBY_REVISION)); /* * The full ruby version string, like ruby -v prints' */ rb_define_global_const("RUBY_DESCRIPTION", MKSTR(description)); /* * The copyright string for ruby */ rb_define_global_const("RUBY_COPYRIGHT", MKSTR(copyright)); /* * The engine or interpreter this ruby uses. */ rb_define_global_const("RUBY_ENGINE", ruby_engine_name = MKSTR(engine)); /* * The version of the engine or interpreter this ruby uses. */ rb_define_global_const("RUBY_ENGINE_VERSION", v); } /*! Prints the version information of the CRuby interpreter to stdout. */ void ruby_show_version(void) { PRINT(description); #ifdef RUBY_LAST_COMMIT_TITLE fputs("last_commit=" RUBY_LAST_COMMIT_TITLE, stdout); #endif #ifdef HAVE_MALLOC_CONF if (malloc_conf) printf("malloc_conf=%s\n", malloc_conf); #endif fflush(stdout); } /*! Prints the copyright notice of the CRuby interpreter to stdout and \em exits * this process successfully. */ void ruby_show_copyright(void) { PRINT(copyright); exit(EXIT_SUCCESS); }


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# A boiler produces 1000 kg/hr steam with rise in enthalpy of steam of 2500 kJ/kg. The calorific value of coal used is 30000 kJ/kg and its consumption is 100 kg/hr. What would be the boiler efficiency?

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1. 18.33%
2. 73.33%
3. 8.33%
4. 83.33%

Option 4 : 83.33%
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## Detailed Solution

Concept:

Boiler efficiency is calculated by the following formula

$$\eta = \frac{{Q ~\times ~\left( {{h_g} - {h_f}} \right)}}{{m ~\times~ GCV\;}}$$

where, Q = steam consumption per hour, hg – hf = rise in enthalpy, m = mass of fuel, GCV = Gross calorific value

Calculation:

Given:

m = 100 kg/hr, CV = 30000 kJ/kg, Q = 1000 kg/hr, Δh = 2500 kJ/kg

$$\eta = \frac{{Q ~\times~ \left( {{h_g} - {h_f}} \right)}}{{m~ \times~ GCV\;}}$$

$$\eta = \frac{{1000 ~\times ~2500}}{{100 ~\times~ 30000}}$$

η = 0.8333 or 83.33%