To fill complete the exercise involving aerobic capacity, you need to complete the exercises and record the time taken for them.
What is aerobic capacity?Aerobic capacity is a term that refers to the sum of the oxygen content of a person's body during a difficult exercise. The aerobic capacity is obtained by multiplying the cardiac output with the oxygen difference in the arterial veins.
So, to complete the time results, you have to engage in an intensive exercise like running a distance and then calculate the amount of oxygen expended during the exercise.
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An Aluminium stick of length 1.5 m is cooled from 200 C to -1800 C. Find the final length if its coefficient of linear expansion is 23×10-6/K?
Answer:
1.454 m
Explanation:
From the question,
L₂ = L₁+α(T₂-T₁).................. Equation 1
Where L₂ = Final Length, L₁ = Initial Length, α = Coefficient of Linear expansion, T₁ = Initial Temperature, T₂ = final Temperature.
Given: L₁ = 1.5 m, α = 23×10⁻⁶ K⁻¹, T₁ = 200 °C, T₂ = -1800 °C
Substitute these values into equation 1
L₂ = 1.5+[23×10⁻⁶ ×( -1800-200)
L₂ = 1.5+(23×10⁻⁶×2000)
L₂ = 1.5+(-0.046)
L₂ = 1.5-0.046
L₂= 1.454 m.
Alejandra is attempting to drag her 38.4-kg Golden Retriever across the wooden floor by applying a horizontal force. What force (in newtons, N) must she apply to move the dog with a constant speed of 0.77 m/s? The coefficient of friction between the dog and the floor is 0.77. (Use the approximation that g ≈ 10 m/s^2)Answer: _________ N
u= coefficient of friction = 0.77
g= gravity = 10 m/s^2
m= mass = 38.4 kg
v= speed = 0.77 m/s
Apply Newton's second law:
Sum of forces = mass x acceleration
Net Force = f - umg
a = 0 (constant speed)
F = umg = 0.77 x 38.4 x 10 = 295.68 N
Carsten is walking down the street while maintaining a constant velocity of 5.0 m/s west. At time t = 0.0 s, Carsten is 250 m west of point A. What is his displacement from the starting position at 60.0 s? *
Answer Format: # units direction
Answer:
550 meters west
Explanation:
Givens: Inital velocity/vi= 5.0m/s, time/t=60.0s, acceleration/a= 0 because a constant velocity means 0 acceleration, inital position/xi=250m
Unknown: final position/xf=?
Formula: xf=xi+vi(t)+1/2at^2
xf=250m+5.0m/s(60.0s)+1/2(0)(60.0s)^2
xf=250m+300m
xf=550m west
its still west because he never chhanged direction therefore his postion kept increasing.
Think about the reasons behind the cost of the Birkin bag and explain why it costs $60,000 in your own words. Do you think that this is a good marketing strategy? Why? Can you think of any other products (clothes / elctronics / anything) that use the same strategy? Does it work?
Answer:
In terms of both price and availability, the Birkin bag is as elusive as it is exclusive. For more than two decades, the bag has been the most sought item in the handbag industry. For such a modest bag, that's a lot of staying power (in looks alone, the design itself takes some 18 hours and is done by a single craftsman).
Of course, it's likely that the bag's rarity is what keeps it at the top of every want list. It is difficult, for example, to stroll into a Hermès store and buy a bag –– even if you could afford to pay the minimum price of $10,000. No, you have to be on an elite list to receive a Birkin.
Explanation:
There are alot more things like the Birkin Bag that use the same marketing skills and it works for alot of companies.
Two trains sound identical horns of frequency 410 Hz. One train is stationary. The other is moving away from an observer, who heats a beat frequency of 35 Hz. How fast is the moving train going?
The speed of the moving train is approximately 33.5 m/s.
The beat frequency is given by the difference in frequency between the two horns, which is equal to the Doppler shift in frequency due to the motion of the moving train. Using the formula for the Doppler effect, we can solve for the speed of the train:
\(f_b = f_s\dfrac{(v + v_o)}{(v + v_s)}\)
where \(f_b\) is the beat frequency, \(f_s\) is the horn frequency, v is the speed of sound, \(v_o\) is the observer's speed, and \(v_s\) is the speed of the source.
We know that \(f_s\) = 410 Hz and \(f_b\) = 35 Hz. The speed of sound in air at standard temperature and pressure is approximately 343 m/s. Since the observer is stationary, \(v_o\) = 0.
Solving for \(v_s\), we get:
\(v_s = \dfrac{(f_s + f_b)}{f_s - 1} \times v\)
\(v_s\) = ((410 Hz + 35 Hz) / 410 Hz - 1) * 343 m/s
\(v_s\) = 33.5 m/s
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(ii) At an instant of time during the oscillations of an L C circuit, the current is momentarily zero. From the same choices, describe the voltage across the capacitor at this instant.(a) It is different from that across the inductor.(b) It is zero. (c) It has its maximum value.(d) It is impossible to determine.
The voltage across the capacitor at the instant when the current is momentarily zero can either have its maximum value (choice c) in a series LC circuit or its minimum value in a parallel LC circuit. Therefore, the correct answer is (c) It has its maximum value.
The voltage across the capacitor in an LC circuit when the current is momentarily zero depends on the specific configuration of the circuit at that instant. To determine the voltage across the capacitor, we need to consider the behavior of the circuit before and after the current becomes zero.
In an LC circuit, the current and voltage oscillate between the capacitor and the inductor. When the current is at its maximum value, the voltage across the capacitor is also at its maximum value. This occurs when the energy stored in the capacitor is maximum and the energy stored in the inductor is zero.
As the current starts decreasing from its maximum value, the voltage across the capacitor starts decreasing as well. When the current becomes zero at a certain instant, the voltage across the capacitor depends on whether the circuit is in a series or parallel configuration.
- In a series LC circuit, the voltage across the capacitor is maximum when the current is zero. This is because the energy is transferred from the inductor to the capacitor, and the capacitor stores the maximum amount of energy.
- In a parallel LC circuit, the voltage across the capacitor is minimum when the current is zero. This is because the energy is transferred from the capacitor to the inductor, and the capacitor stores the minimum amount of energy.
So, the voltage across the capacitor at the instant when the current is momentarily zero can either have its maximum value (choice c) in a series LC circuit or its minimum value in a parallel LC circuit. Therefore, the correct answer is (c) It has its maximum value.
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A stone was dropped off a cliff and hit the ground with a speed of 112 ft/s. What is the height (in feet) of the cliff? (use 32 ft/s2 for the acceleration due to gravity. ).
The height of the cliff from which the stone was dropped is equal to 196 ft.
What is the equation of motion?The equations of motion can be applied to establish the relationship between the velocity (v and u), acceleration (a), time, and displacement (s) of a moving object.
The mathematical representations of the equations of motions are:
\(v= u+ at\\S = ut+\frac{1}{2}at^2\\ v^2-u^2=2aS\)
Given, the final speed of the stone, v = 112 ft/s
The initial speed of the stone will be, u = 0
The acceleration due to gravity, a = 32 ft/s²
From the equation of motion, find the time taken by the stone to reach the ground:
v = u + at
112 = 0 + (32) t
t = 3.5 sec
The equation of motion can be used to find the height of the cliff:
H = ut + (1/2)gt²
H = 0 + (1/2)× 32×(3.5)²
H = 196 ft
Therefore, the height of the cliff is 196 feet.
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A pitcher threw a baseball straight up at 35.8 meters per second. What was the ball's velocity after 2.50 seconds? (Note that, although the baseball is still climbing, gravity is accelerating it downward.)
Answer:
The velocity after 2.5 seconds is 11.3 m/s
Explanation:
Given;
u = 35.8 m/s
t = 2.50s
g = 9.8 m/s^2
Mathematically;
V = U-gt
Substituting these values;
V = 35.8-2.5(9.8)
V = 35.8-24.5
V = 11.3 m/s
The kinetic energy of an object increases as its A height B. speed increases C. volume D. potential energy
Answer:
As its speed increases.......
The kinetic energy of an object increases as its speed increases. So, option B.
What is meant by kinetic energy ?The type of energy that an object possesses as a result of its motion is called kinetic energy.
The kinetic energy of an object is described as the amount of effort required to accelerate a body of a specific mass from rest to its specified velocity.
The body keeps its kinetic energy after gaining it during acceleration, barring changes in speed.
When slowing down from its current pace to a condition of rest, the body expends the same amount of energy.
A force must be applied to an object in order to accelerate it. We must put in effort in order to apply a force. After the work is finished, energy is transferred to the item, which then moves at a new, constant speed.
The kinetic energy of the object,
KE = 1/mv²
where m is the mass and v is the speed of the object.
Since, the mass is a constant, the increase in speed can increase the kinetic energy of the object.
Hence,
The kinetic energy of an object increases as its speed increases.
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what equation describes these relationships?
1. (-3,-9), (-2,-6), (-1,-3), (0,0)
2. (3,0), (2,-1), (1,-2), (0,-3)
3. (0,0), (1,-3), (2,-6), (3,-9)
a. y=3x
b. y=-3x
c. y=x-3
Can someone please help
Answer:
55 km
pls give brainliest hope it help
Explanation:
The vertical columns on the periodic table of elements are called groups.
O A. True
B. False
Answer:
A. True
Explanation:
The vertical columns on the periodic table are called groups or families because of their similar chemical behavior.
Brainlist Pls!
17. For how long should a force of 130 N be applied to an object of mass 50 kg to change its speed from 20 m/s to 60 m/s?
a. 0.031 s
b. 0.065 s
c. 15.4 s
d. 40 s
Answer:
c. 15.4 s
Explanation:
Given the following data;
Mass, m = 50kg
Force, F = 130N
Initial velocity, u = 20m/s
Final velocity, v = 60m/s
To find the time;
First of all, we would solve for acceleration using the formula below;
Force = mass * acceleration
130 = 50*acceleration
Acceleration = 130/50
Acceleration = 2.6m/s²
Now, we would use the first equation of motion to find the time.
V = U + at
60 = 20 + 2.6t
2.6t = 60 - 20
2.6t = 40
t = 40/2.6
Time, t = 15.39 ≈ 15.4 seconds.
The time taken will be 15.4 seconds approximately. The correct answer is option c
NEWTON SECOND LAW
Newton's second law state that, the rate of change of momentum is proportional to the force applied.
Given that a force of 130 N be applied to an object of mass 50 kg to change its speed from 20 m/s to 60 m/s. The given parameters are;
Force F = 130 Nmass m = 50 kgInitial velocity u = 20 m/sFinal velocity V = 60 m/sfrom Newton's second law,
Ft = m(v - u)
Substitute all the parameters into the formula
130t = 50 (60 - 20)
130t = 50 x 40
130t = 2000
t = 2000/130
t = 15.38 seconds
Therefore, the time taken will be 15.4 seconds approximately. The correct answer is option c
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The average mass of an adult American female is 75 kilograms. What is the weight of
the average female in pounds?
Answer:
with 75kg --> 165.35 pound(lb)
Explanation:
\( \frac{75}{0.45359237} \)
A boy wishes to make a catapult out of a rubber band of width 9mm and thickness 1.55mm. Determine the length of the band that must use so that when he stretches it by 0.25 of its natural length and releases it the velocity of a pebble of mass 0.006kg will be 30m\s. take the young modulus of the rubber to be 4×10^7 N\m^2
Answer:
To solve this problem, we can use the formula for the potential energy stored in a stretched spring or rubber band: U = (1/2) k x^2
where U is the potential energy, k is the spring constant, and x is the amount of stretch.
We can rearrange this formula to solve for k: k = 2 U / x^2
The velocity of the pebble can be found using conservation of energy:
(1/2) m v^2 = U
where m is the mass of the pebble and v is its velocity.
Rearranging this formula, we get: v = sqrt(2 U / m)
We can combine these formulas to solve for the length of the rubber band:
k = (4 U) / (0.25 L^2)
v = sqrt((8 U) / (0.006))
where L is the original length of the rubber band.
Since the width and thickness of the rubber band are given, we can calculate its cross-sectional area:
A = (9 mm) x (1.55 mm) = 13.95 mm^2 = 1.395 x 10^-5 m^2
Using the Young's modulus given in the problem, we can calculate the spring constant: k = (A / L) x (Y / 4)
where Y is the Young's modulus.
The formula for k above, we get: (4 A Y / L^3) x (U / 0.25) = 0.006 v^2
Solving for L, we get: L = (4 A Y U / 0.006 v^2)^1/3
Substituting the given values and solving, we get: L = 34.86 cm
Therefore, the length of the rubber band should be approximately 34.86 cm to achieve the desired velocity of the pebble
According to Newton’s law of universal gravitation, which statements are true?
A: As we move to higher altitudes, the force of gravity on us decreases.
B: As we move to higher altitudes, the force of gravity on us increases.
C: As we gain mass, the force of gravity on us decreases.
D: As we gain mass, the force of gravity on us increases.
E: As we move faster, the force of gravity on us increases.
Before we solve this, we should know this fact:
According to Newton's Law of Gravitation, the force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining the centres of the two objects. It can be shown by this:
\(F ∝ \frac{Mm}{ {d}^{2} } \)
Now, let us check all the options.
A. As we move to higher altitudes, the force of gravity on us decreases.
This statement is true.
The force of gravity is inversely proportional to the square of distance from the centre of the earth. If, we go up the surface of the earth, the distance from the centre of the earth increases and hence the value of force of gravity decrease. So, force of gravity decreases with altitude.
B. As we move to higher altitudes, the force of gravity on us increases.
This statement is false.
We have already got the result in option A. that the force of gravity decreases with altitude. It never increases with altitude.
C. As we gain mass, the force of gravity on us decreases.
This statement is false.
The force of gravity is directly proportional to the product of the masses. So, if increase our mass, then the force of gravity will also increase and if we decrease our mass, then the force of gravity decreases.
D. As we gain mass, the force of gravity on us increases.
This statement is true.
As mentioned earlier in option C., the force of gravity is directly proportional to the product of the masses of the earth and another object. So, as we gain mass, the force of gravity on us increases.
E. As we move faster, the force of gravity on us increases.
This statement is true.
Here, we have to consider a different formula. According to Newton's Second Law,
F = ma, where F is the force, m is the mass and a is the acceleration.
In other words,
F ∝ a, i.e., force is directly proportional to acceleration.
We know, acceleration is the rate of change of velocity of an body within a time period.
So, if speed is increased, then acceleration will also be greater, which results in the increase of force. So, as we move faster, the force of gravity on us increases.
Answers:
A: As we move to higher altitudes, the force of gravity on us decreases.
D: As we gain mass, the force of gravity on us increases.
E: As we move faster, the force of gravity on us increases.
Hope you could understand.
If you have any query, feel free to ask.
What is the momentum of a 9,800 kg bus travelling at 40 mph in ? (kg*m/s)
Given data,
Mass of the bus is m = 9800 kg,
Speed of the bus is v = 40 mph,
Converting speed of the bus in m/s, {1 miles per hour = 0.447 meter per second}
v = 40*0.447
v = 17.88
Calculating momentum of the bus,
\(\begin{gathered} P=\text{ mv} \\ P=\text{ 9800}\times17.88 \\ P=\text{ 175,224 kg.m/s} \end{gathered}\)genevieve works for a company that manufactures office furniture. it is her responsibility to source component parts such as casters, laminate, and plastic to make the office furniture. genevieve works in blank . multiple choice question. operations r
To source component parts such as casters, laminate, and plastic to make the office furniture, genevieve works in d. resources.
A furniture maker's hallmark is a manufacturing tag, label, or stamp with the name of the maker. Such marks and labels may have been on the drawers of old dressers, the back of dressers, or under the seats of chairs and sofas.
His four types of manufactures office furniture that are common in all types of workspaces include desks, chairs, tables and storage units. While desks are primarily intended for employee workstations, chairs can be used for both employees and customers/suppliers/customers.
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Disclaimer:- your question is incomplete, please see below for the comlete question.
a. outcomes.
b. products
c. services.
d. resources.
e. output management
UNIT (5) Momentum 5.D Change in Momentum DATE NAME Scenario A 10 kg box, initially at rest, moves along a smooth horizontal surface. A horizontal force is applied to the box. The magnitude of the force changes as a function of time as shouwn. Take the positive direction to be to the right. Tme (econd Data Analysis Rank the magnitude of the impulse applied to the box by the force during each 2-second interval indicated below: PART A: A. 0-2 seconds B. 24 seconds C. 4-6 seconds D.6-8 seconds E.8-10 seconds Smallest impulse Greatest impulse Write a few sentences justifying your reasoning. Use words like speed, velocity, acceleration, time, force, momentum, and impulse. PART B: Using Representations PART C: Re-represent the data given in the force vs. time graph in Part A as a momentum vs. time graph for the same 10 kg box. Step 1: Identify the equation that relates force and momentum. Step 2: How can momentum be found from a force vs. time graph? Step 3: Plot the momentum as a function of time. (Make a table if you need to.) 10Time (s) Momentum (kgm/s) Forte Newtons) 5.D Change in Momentum Quantitative Analysis Use the graph above to calculate the velocity of the box after 10 seconds. For each line in the calculation, explain what was done mathematically. The first line is done for you. Ap = mai The change in momentum is equal to the mass times the change in velocity. UNIT 5. Momentum 5.D Change in Momentum NAME DATE Scenario A 10 kg box, initially at rest, moves along a smooth horizontal surface. A horizontal force is opplied to the box. The magnitude of the force changes as a function of time as shoun. Take the positive direction to be to the right. Tme econd Data Analysis Rank the magnitude of the impulse applied to the box by the force during each 2-second interval indicated below: PART A A. 0-2 seconds B. 2-4 seconds C. 4-6 seconds D. 6-8seconds E.8-10 seconds Greatest impulse Smallest impulse Write a few sentences justifying your reasoning. Use words like speed, velocity, acceleration, time, force, momentum, and impulse. PART B: Using Representations PART C: Re-represent the data given in the force vs. time graph in Part A as a momentum vs. time graph for the same 10 kg box. Step 1: Identify the equation that relates force and momentum. Step 2 How can momentum be found from a force vs. time graph? Step 3: Plot the momentum as a funetion of time. (Make a table if you need to) 10 Time (s) | 135 Return to Table of Contents 2Categeard Momentum (kom/s) 5.D Change in Momentum Quantitative Analysis Use the graph above to calculate the velocity of the bax after 10 seconds. For each line in the calculation, explain what was done mathematically. The first line is done for you. Ap = mAi The change in momentumis equal to the mass times the change in velocity.
From the given data,
Part A: -In opposite direction Greatest impulse B>C>A>D>E smallest impulse
Part B: -increasing constantly thus acceleration is increasing resulting in increase of velocity and momentum as well.
Part C: -\(Force $=m$ a $\) =\($m \frac{\Delta v}{\Delta t}=\) \(m$ momentum\)
\(F=\frac{m}{\Delta t}\)
Mass of box \(=10kg\)
Impulse = \(F_{average}\) At
A. 0-2 seconds
Impulse = 1.5×2 = 3N sec
B. 2-4 sec
Impulse = 3×2 = 6N sec
C. 4-6 sec
Impulse = 2.75×2 =5.5N sec
D. 6-8 sec
Impulse = 1×2 = 2N sec
E. 8-10 sec
Impulse = 1×2 = 2N sec
In opposite direction Greatest impulse B>C>A>D>E smallest impulse.
For the first two seconds force is increasing constantly thus acceleration is increasing resulting in increase of velocity and momentum as well. Now for the next three seconds a constant force is acting on the box resulting in constant acceleration. After that for the next three seconds force goes on decreasing constantly thus deacceleration and decrease of impulse and velocity. In last 2 seconds deacceleration is in opposite direction that is how box starts moving in opposite direction
\((1) Force $=m$ a $\)\($=m \frac{\Delta v}{\Delta t}=\)\(m$ momentum\)
\(F=\frac{m}{\Delta t}\)
(2) Area under the curve gives the momentum
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Convert this decimal into its fractional
form simplified completely.
0.350
Answer:
7/20
Explanation:
0.35 as a fraction =35/100
35/100 can b divided by 5 so =7/20
0.350 simplified as a fraction is 7/20
Answer:
7/20
Explanation:
Mass can be measured with a triple
beam balance in units of
Resources
A. grams
B. Newtons
C. cubic centimeters
D. millimeters
Answer:
A
Explanation:
A net force of 5 N was used to push a 4 kg crate. By how much did the crate accelerate?
Answer:
125 m/s^2
Explanation:
a tightly wound toroid of inner radius 1.3 cm and outer radius 2.6 cm has 1080 turns of wire and carries a current of 2.1 a. what is the magnetic field a distance 1.95 cm from the center?
At 0.9 cm from the toroid's center, there is a 0.053 T magnetic field present.
the details provided;
The toroid's inner radius is 1.3 cm (0.013 m) and
its outer radius is 2.6 cm (0.026 m).
There are 1080 revolutions in the toroid.
carries a current of 2.1.
The following formula is used to compute the magnetic field at a distance of 1.95 cm from the toroid's center:
B= (μNI)/2\(\pi\)R
B=(4\(\pi\)*\(10^{-7}\))*(1080)*(2.1)/2\(\pi\)*0.0195
B=0.023 T.
Simply put, what is a magnetic field?The magnetic field is the area in which a magnet produces its magnetic effects. To describe how the magnetic force is dispersed in the area around and inside something magnetic in nature, we utilize the magnetic field as a tool.
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What happens to a hockey puck when a net force of 5 N acts on it? a It accelerates. b It remains at rest. c It increases in mass. d It moves at a constant speed.
Answer:
I think it is A
Explanation:
A gas is compressed and does P-V work on the system equal to 280 J. At the same time, it releases 216 J of heat to the surroundings. What is the change in energy of the system
The change in energy of the system is negative 496 J.
The change in energy of the system can be calculated using the first law of thermodynamics, which states that the change in energy of a system is equal to the heat added to the system minus the work done by the system:
ΔE = Q - W
Where:
ΔE is the change in energy of the system
Q is the heat added to the system
W is the work done by the system
In this case, the gas does P-V work on the system, which is positive since work is being done on the system. Therefore, W = 280 J.
The gas also releases heat to the surroundings, which is negative since heat is leaving the system. Therefore, Q = -216 J.
Substituting these values into the equation, we have:
ΔE = -216 J - 280 J
ΔE = -496 J
The change in energy of the system is -496 J. The negative sign indicates that the system has lost energy.
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calculate the force applied to raise a table to a height of 8m if 120j of work was done
The force applied to raise a table to a height of 8m when 120 Joules of work is done is 15 N. That is, Force F = 15N
What is Work ?Work is the product of force and distance in the direction of force. Work is done on an object when the displacement of the object is in the direction of the force applied. That is,
W = F x S
The S.I unit of work is Joule.
Given that a table to a height of 8m and 120J of work was done. To find the force applies, the parameters to consider are
Work W = 120 JDistance S = 8 mForce F = ?Make force F the subject of formula from the formula written above. That is,
F = W/S
F = 120 / 8
F = 15 N
Therefore, if the 120 Joule of work is done, the force applied to raise a table to a height of 8m is 15 Newtons.
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what do our ears do
Answer:
hear
Explanation:
stuff stuff moaning and stuff
Can someone help me please this test is very important
Answer:
Friction is caused in roller coasters by the rubbing of the car wheels on the track and by the rubbing of air (and sometimes water!) against the cars.
Explanation:
Use Q=mCΔt : PCl3 is a compound used to manufacture pesticides. A reaction requires that 96.7 g of PCl3 be raised from 31.7 oC to 69.2 oC. How much energy will this require given that the specific heat of PCl3 is 0.874 J/g oC? Please explain fully . Im trying to understand this
The quantity of heat required is 1229.1 J
What is specific heat capacity?The specific heat capacity of a substance is the quantity of heat required to increase the temperature of a unit mass of the substance by 1°C or 1k. it measure in J/kg/k.
The specific heat capacity of a substance is denoted by 'C'. And it is expressed as C = Q/m∆t
where ∆t is the change in temperature, Q is the quantity of heat energy required and m is the mass of the substance.
therefore Q=mCΔt
Q= 96.7× 0.874×(69.2-31.7)
Q= 1229.1J
Thereore the heat required to increase the temperature from 31.7°C to 69.2°C is 1229.1J
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Use Gaussian elimination to solve the systems: (a) 2x - 2y-z = -2 4x + y = 2z = 1 -2x+y=z=-3 (b) x+2y=z=2 2x-y+z=2 3y+z=4 (c) 2x + y - 4z=-7 x-y+z=-2 -x+3y-2z = 6
a) z = (-6) / (-24/5) = 5/2
y = (5 - 4z) / 5 = -1/2
x = (-2 + z - y) / 2 = 1/2
b) z = (2/5) / (-9/5) = -2/9
y = (-2 - z) / -5 = 2/5
x = (2 - 2y - z) / 1 = 4/9
c) x = t
y = (1 + t) / 3
z = t
(a) To solve the system of equations using Gaussian elimination:
1. Write the augmented matrix:
[2 -2 -1 | -2]
[4 1 2 | 1]
[-2 1 -3 | -3]
2. Apply row operations to transform the matrix into row-echelon form:
R2 = R2 - 2R1
R3 = R3 + R1
The resulting matrix is:
[2 -2 -1 | -2]
[0 5 4 | 5]
[0 1 -4 | -5]
3. Further row operations:
R3 = R3 - (1/5)R2
The matrix becomes:
[2 -2 -1 | -2]
[0 5 4 | 5]
[0 0 -24/5 | -6]
4. Solve for the variables using back substitution:
z = (-6) / (-24/5) = 5/2
y = (5 - 4z) / 5 = -1/2
x = (-2 + z - y) / 2 = 1/2
(b) To solve the system of equations using Gaussian elimination:
1. Write the augmented matrix:
[1 2 1 | 2]
[2 -1 1 | 2]
[0 3 1 | 4]
2. Apply row operations to achieve row-echelon form:
R2 = R2 - 2R1
R3 = R3 - 2R1
The resulting matrix is:
[1 2 1 | 2]
[0 -5 -1 | -2]
[0 -1 -1 | 0]
3. Further row operations:
R3 = R3 - (1/5)R2
The matrix becomes:
[1 2 1 | 2]
[0 -5 -1 | -2]
[0 0 -9/5 | 2/5]
4. Solve for the variables using back substitution:
z = (2/5) / (-9/5) = -2/9
y = (-2 - z) / -5 = 2/5
x = (2 - 2y - z) / 1 = 4/9
(c) To solve the system of equations using Gaussian elimination:
1. Write the augmented matrix:
[2 1 -4 | -7]
[1 -1 1 | -2]
[-1 3 -2 | 6]
2. Apply row operations to obtain row-echelon form:
R2 = R2 - (1/2)R1
R3 = R3 + R1
The resulting matrix is:
[2 1 -4 | -7]
[0 -3 3 | 1]
[0 4 -6 | -1]
3. Further row operations:
R3 = R3 + (4/3)R2
The matrix becomes:
[2 1 -4 | -7]
[0 -3 3 | 1]
[0 0 0 | 0]
4. Solve for the variables using back substitution:
Let's denote a free variable as t.
x = t
y = (1 + t) / 3
z = t
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To solve the system of equations, we can use Gaussian elimination and convert the equations to an augmented matrix. However, in this case, the row-echelon form shows that the system is inconsistent and has no solution.
Explanation:To solve the system of equations using Gaussian elimination, we can use the augmented matrix. First we convert the system of equations into augmented matrix form:
2 -2 -1 -2Now, we perform row operations to obtain the row-echelon form:
1 -1/2 -1/2 -1From the row-echelon form, we can see that the system of equations is inconsistent as the last equation is always satisfied. Therefore, there is no solution for this system.
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