The data listed below are claimed for a power cycle operating between hot and cold reservoirs at 1000 K and 300 K, respectively. For each case, determine whether the cycle operates reversibly, operates irreversibly, or is impossible.
a) QH = 600 kJ, Wcycle = 300 kJ, QC = 300 kJ
b) QH = 400 kJ, Wcycle = 280 kJ, QC = 120 kJ
c) QH = 700 kJ, Wcycle = 300 kJ, QC = 500 kJ
d) QH = 800 kJ, Wcycle = 600 kJ, QC = 200 kJ

Answers

Answer 1

To determine whether the power cycle operates reversibly, irreversibly, or is impossible, we will use the following relationships:
1. The efficiency of a reversible cycle is given by η_rev = 1 - (T_C / T_H)
2. The efficiency of an irreversible cycle is given by η_irr = W_ cycle / Q_H
3. If η_irr > η_rev, the cycle is impossible.

Explanation:

For each case, we will calculate the reversible efficiency and compare it to the irreversible efficiency:

a) QH = 600 kJ, Wcycle = 300 kJ, QC = 300 kJ
η_rev = 1 - (300 / 1000) = 0.7
η_irr = 300 / 600 = 0.5
Since η_irr < η_rev, the cycle operates irreversibly.

b) QH = 400 kJ, Wcycle = 280 kJ, QC = 120 kJ
η_rev = 1 - (300 / 1000) = 0.7
η_irr = 280 / 400 = 0.7
Since η_irr = η_rev, the cycle operates reversibly.

c) QH = 700 kJ, Wcycle = 300 kJ, QC = 500 kJ
η_rev = 1 - (300 / 1000) = 0.7
η_irr = 300 / 700 ≈ 0.429
Since η_irr < η_rev, the cycle operates irreversibly.

d) QH = 800 kJ, Wcycle = 600 kJ, QC = 200 kJ
η_rev = 1 - (300 / 1000) = 0.7
η_irr = 600 / 800 = 0.75
Since η_irr > η_rev, the cycle is impossible.

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Answer 2

To determine whether each power cycle operates reversibly, irreversibly, or is impossible, we can use the following equation for the thermal efficiency of a Carnot cycle:

η = 1 - QC/QH

If the actual cycle has a thermal efficiency less than the Carnot cycle, then it must be operating irreversibly. If the actual cycle has a thermal efficiency equal to the Carnot cycle, then it must be operating reversibly. If the actual cycle has a thermal efficiency greater than the Carnot cycle, then it is impossible.

a) QH = 600 kJ, Wcycle = 300 kJ, QC = 300 kJ

The thermal efficiency of the Carnot cycle is:

η = 1 - QC/QH

η = 1 - 300/600

η = 0.5

The actual cycle's thermal efficiency is:

η = Wcycle/QH

η = 300/600

η = 0.5

Since the actual cycle's thermal efficiency is equal to the Carnot cycle's thermal efficiency, the cycle operates reversibly.

b) QH = 400 kJ, Wcycle = 280 kJ, QC = 120 kJ

The thermal efficiency of the Carnot cycle is:

η = 1 - QC/QH

η = 1 - 120/400

η = 0.7

The actual cycle's thermal efficiency is:

η = Wcycle/QH

η = 280/400

η = 0.7

Since the actual cycle's thermal efficiency is equal to the Carnot cycle's thermal efficiency, the cycle operates reversibly.

c) QH = 700 kJ, Wcycle = 300 kJ, QC = 500 kJ

The thermal efficiency of the Carnot cycle is:

η = 1 - QC/QH

η = 1 - 500/700

η = 0.286

The actual cycle's thermal efficiency is:

η = Wcycle/QH

η = 300/700

η = 0.429

Since the actual cycle's thermal efficiency is greater than the Carnot cycle's thermal efficiency, the cycle is impossible.

d) QH = 800 kJ, Wcycle = 600 kJ, QC = 200 kJ

The thermal efficiency of the Carnot cycle is:

η = 1 - QC/QH

η = 1 - 200/800

η = 0.75

The actual cycle's thermal efficiency is:

η = Wcycle/QH

η = 600/800

η = 0.75

Since the actual cycle's thermal efficiency is equal to the Carnot cycle's thermal efficiency, the cycle operates reversibly.

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Related Questions

How do you line up vectors so you can add them?

Answers

Answer:

The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.

Explanation:

two vectors, you place the tail of the second vector to the tip of the first vector while maintaining vector orientation. The resultant vector is a vector that begins at the tail of the first vector and points in a straight line to the tip of the second vector.

I need the answer for both questions plzZz

I need the answer for both questions plzZz

Answers

Answer:

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Explanation:

Answer:

i dont even know

Explanation:

I just dont know

Calculate the kinetic energy of a tennis ball that has a mass of .056 kg and is moving at a speed of 44 m/s

Answers

\(\text{Kinetic energy,}\\\\E_k = \dfrac 12 mv^2\\\\~~~~~=\dfrac 12 \times 0.056 \times 44^2\\\\~~~~~=54.208~J\)

A student swings ball of mass M on the end of a string in vertical circle of radius R,as shown in the figure below. Also shown is diagram representing all the forces exerted on the ball at the bottom of the circle where its speed is What is the magnitude of the acceleration of the ball at the bottom of the circle? FTension FGravity
A)Fi /M
B)Fc /M
C)Fr+Fg/M
D) Ft- Fg/M

Answers

The magnitude of the acceleration of the ball  applied at the bottom of the circle is expressed as FTension-FGravity/M.

When making a vertical circle with a ball on a string?The string's tension varies along its vertical as well as  circular course. The ball's speed varies as long as the total amount of kinetic and potential energy is constant through out. Centripetal force fluctuates when speed changes accordingly.

How can I determine the tension of a string moving in a vertical circle?

You can write it down as FC = mv2 /r.

The centripetal force in a moving object attached to a string is equal to the product of the object's mass (mg) and the string tension (T).

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Use the universal law of gravitation to solve the following problem.

The force of gravity between Jake and his daughter Annika is 1.67 × 10–9 N. If they are 11 m apart and Jake has a mass of 110 kg, what is Annika's mass?

a. Write out the formula for this problem.

b. Plug in the values from this problem into the formula.

c. Solve the problem, writing out each step.

d. Correct answer

Answers

Answer:

Explanation:

Formula

Force = G * m1 * m2 / r^2

Givens

F = 1.67 * 10 - 9 NG = 6.67 × 10-11 Newtons kg-2 m2.m1 = 110 kgm2 = ?r = 11 meters.

Solution

1.67 * 10^-9 = 6.67 * 10^-11 * 110 * m2/(11)^2        Divide both sides by 10^-9

1.67 * 10^-9/10^-9 = 6.67*10^-11/(10^-9 *110*m2/121

1.67 = 6.67 * 110 * 10^-2 * m2 / 121                      Multiply both sides by 121

1.67 * 121  = 6.67 * 10 ^-2 * 110 * m2 * 121/121      Combine

202.07 = 7.337 * m2                                           Divide both sides by 7.337

202.07/7.337  =  m2

m2 = 27.54 kg

Please help, quicklyyyyy

Please help, quicklyyyyy

Answers

1 because I’m going out of town to see the

Find the average distance (in the Earth's frame of reference) covered by the muons if their speed relative to Earth is 0. 845 c. Note: the rest lifetime of a muon is 2. 2 10's Consider muons traveling toward Earth from their point of creation at a height of 5. 00 km. Express your answer to three significant figures

Answers

The average distance travelled by the muons is 5.50 km in the Earth's frame of reference.

Muon is a subatomic particle that is a fundamental constituent of matter. It is classified as a lepton, along with the electron, tau, and three neutrinos. A muon's rest mass is 105.65837 MeV/c², which is around 207 times greater than the electron's rest mass. A muon's rest lifetime is 2.2 microseconds.

Find the average distance covered by the muons if their speed relative to Earth is 0.845c. The muon's lifetime can be used to determine the average distance it travels if its speed is constant over that time. The distance can be calculated using the following formula:

Distance = Speed × Time

A muon's lifetime of 2.2 microseconds and a relative velocity of 0.845c are given. We can use the above formula to determine the average distance covered by a muon in this situation.

Distance = Speed × Time= 0.845c × 2.2 µs= 4.97 × 10⁻⁴ km or 497 meters.

Since the muons are travelling towards Earth from a height of 5.00 km, we can add the height of their point of creation to the distance they travelled to determine the average distance they travelled from creation to the Earth's surface.

Average distance travelled by muon = Distance + Height= 497 m + 5.00 km= 5.50 km (to 3 significant figures).

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is a
The wire in a simple circuit
which forms an
unbroken pathway, allowing the
electric charges to flow.
A. switch
B. insulator
C. resistor
D. conductor

is aThe wire in a simple circuitwhich forms anunbroken pathway, allowing theelectric charges to flow.A.

Answers

Answer:

I believe the answer is D: Conductor.

a tortoise and a hare cover the same distance in a race. the hare goes very fast for brief intervals, but stops frequently, whereas the tortoise plods along steadily and finishes the race ahead of the hare.which of the two racers is likely to reach the greatest instantaneous speed during the race?

Answers

The hare has the greater instantaneous speed since he can go faster.

In a race, both the turtle and the hare travel the same distance. The tortoise travels gradually and finishes the race before the hare, whereas the hare travels very quickly for brief periods of time but frequently stops.

Since he can move more quickly, the hare has a faster instantaneous speed. Since the turtle won, he moves at a faster average speed.

An object's instantaneous speed is a measure of how quickly it is travelling right now. The speed and direction of an object's motion are both included in its instantaneous velocity, which is a vector quantity.

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A2.0kg arrow is fired at 10m/s into a stationary 10 kg block of ice as shown below: the arrow sticks into the block of ice and they start moving . What is the velocity of the arrow/ice combination after the collison

Answers

Answer:

the velocity is 1.67 m/s

Explanation:

The computation of the velocity of the arrow/ice combination after the collison is shown below:

Here the law of conservation is applied

\(p_i = p_f\\\\m_av_a+m_b+v_b= (m_a + m_b)v\)

Here

\(m_a = 2.0 kg\\\\v_a = 10m/s\\\\m_b = 10 kg\\\\v_b = 0\)

Now

\(v = \frac{m_av_a+ m_bv_b}{m_a + m_b} \\\\= \frac{(2.0kg) (10 m/s) + 0}{2.0kg + 10.0kg}\)

= 1.67 m/s

hence, the velocity is 1.67 m/s

Rank the following configurations according to the magnitude of the force felt at point P

— strongest force = top
— weakest force = bottom

**20 points!!**

Spam/unrelated content= reporting

Rank the following configurations according to the magnitude of the force felt at point P strongest force

Answers

Answer:

3, 4, 1, 2

Explanation:

Two forces act on a 39 kg mass to give it an acceleration of 40 m/s 2 in the positive x direction. If one of the forces acts in the negative y direction with a magnitude of 150 N, what is the magnitude of the second force? Answer in units of N.

Answers

The magnitude of the second force is 1710 N.

What is force?

Force is the product of mass and acceleration.

To calculate the magnitude of the second force, we use the formula below

Formula:

F' = F+ma................... Equation 1

Where:

F' = Magnitude of the second forceF = Magnitude of the first forcem = Mass a = Acceleration

From the question,

Given:

F = 150 Nm = 39 kga = 40 m/s²

Substitute these values into equation 1

F' = 150+(40×39)F' = 150+1560F' = 1710 N

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a piece of 14-gauge copper wire (meaning that it has a diameter of 1.63mm) has a resistance of 0.128 . the resistivity of copper is . what must be the length of the wire?

Answers

The length of the copper wire must be approximately 44.2 meters for it to have a resistance of 0.128 ohms, assuming a 14-gauge wire with a diameter of 1.63 mm and using the resistivity of copper.

To find the length of the wire, we can use Ohm's Law, which states that the resistance (R) is equal to the product of the resistivity (ρ), the length (L), and the cross-sectional area (A) of the wire, divided by the diameter (d) of the wire squared.

The formula can be written as:

R = ρ * (L / A)

Resistance (R) = 0.128 ohms

Resistivity of copper (ρ) = (1.68 × 10^-8) ohm-meter (at 20°C)

Diameter (d) = 1.63 mm = 0.00163 meters (converted from millimeters to meters)

We need to find the length (L) of the wire.

To calculate the cross-sectional area (A) of the wire, we can use the formula for the area of a circle:

A = π * (d/2)^2

Plugging in the values, we have:

A = 3.14159 * (0.00163 / 2)^2

A  ≈ 2.08 x 10^-6 square meters

Rearranging Ohm's Law to solve for the length (L), we get:

L = (R * A) / ρ

Substituting the given values:

L = (0.128 * 2.08 x 10^-6) / (1.68 x 10^-8)

L  ≈ 44.2 meters

The length of the copper wire must be approximately 44.2 meters for it to have a resistance of 0.128 ohms, assuming a 14-gauge wire with a diameter of 1.63 mm and using the resistivity of copper.

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WILL GIVE BRAINLIEST.
Objects 1 and 2 attract each other with a gravitational force of 18.0 units. If the distance separating Objects 1 and 2 is doubled, then the new gravitational force will be ____ units.

Answers

If the distance separating Objects 1 and 2 is doubled, then the new gravitational force will be 72.0 units.

What is Gravitational force?

The force of attraction that operates on all matter is known as gravity, or gravitational force in mechanics. It is by far the weakest known force in nature and plays no role in determining the internal properties of everyday matter. It controls the orbits of objects in places, the structure, and the evolution of stars, galaxies, and the universe as a whole.

On Earth, every object has a weight, or downward gravitational force, proportional to its mass, and the mass of the Earth acts on them. The acceleration that gravity causes in a falling item serves as a gauge of its strength. The acceleration of gravity at the Earth's surface is around 9.8 meters per second (32 feet) per second. For each second her body is in free fall, its velocity increases by about 9.8 meters per second. On the moon, the acceleration of a free-falling object is approximately 1.6 meters per second.

The gravitational force among two objects can be written as:

F = G×m₁×m₂/r²

Where G is the gravitational constant, m₁ and m₂ are the masses of both objects and r is the distance between them.

Now, we have that F = 18 .0 units.

Let's change r to 2r, the new force is:

F' = G×m₁×m₂/(2r)²

F' = G×m₁×m₂/4r²

F'/F = (G×m₁×m₂/4r²)/G×m₁×m₂/r²

F' = 4 F

F' = 4 × 18.0

F' = 72 units

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If the electric field between the plates of a given air-filled capacitor is weakened by removing charge from the plates, the capacitance of that capacitor
A) increases.
B) decreases.
C) does not change.
D) It cannot be determined from the information given.

Answers

The capacitance of an air-filled capacitors does not change is if electric field in between plates is made weaker by discharging the plates:

How does a weak capacitor affect things?

The external unit's malfunctioning due to a defective capacitor impedes the cooling procedure as a whole. Second, the system must work harder to complete its task due to poor voltage distribution to outer unit components. A defective capacitor frequently causes harm to additional components.

What is the electric field that exists between two capacitor plates?

In a simple parallel-plate capacitor, applying a voltage between 2 conducting plates creates an equal electric field between both the plates. The electrical field intensity in a capacitor is inversely related to the distance between the plates but directly proportional to the applied voltage.

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which electromagnetic radiation wavelength is efficiently observed using ground based telescopes?

Answers

Visible radiations are the electromagnetic radiation wavelength which is efficiently observed using ground based telescopes.

The electromagnetic waves we can for the most part see on the ground comprise of apparent light, which is challenging for the air to retain, close infrared beams, and a few electromagnetic waves. These frequency ranges are called air windows.

Noticeable light is a sort of electromagnetic radiation, which is sent in waves or particles at various frequencies and frequencies. This wide scope of frequencies is known as the electromagnetic range. That range is commonly separated into seven districts arranged by diminishing frequency and expanding energy and frequency.

A commonplace natural eye will answer frequencies from around 380 to around 750 nanometers. As far as recurrence, this compares to a band nearby 400-790 terahertz. These limits are not pointedly characterized and may differ per person. Under ideal circumstances these restrictions of human insight can stretch out to 310 nm (bright) and 1100 nm (close to infrared).

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"Please include all relevant working out as detailed as possible
and all relevant diagram to find the answer. Much appreciated! I
will upvote! Thank you so much"
Find total response of the system (transient+steady state). Do not solve for coefficients. Determine the frequency of applied force at which resonance will occur? M = 20 kg F, = 90 N Given: -6 rad/s M

Answers

Given the following information:Mass of the system, m = 20 kg.Damping coefficient, b = 6 Ns/m.Force, F = 90 N.Frequency of applied force, f = ?Applied force angular frequency, w = 6 rad/s.Forced vibration equation:F(t) = F0 sin(wt)where F0 = 90 N and w = 6 rad/s.Under the action of the force F, the mass m will oscillate.The equation of motion for the mass-spring-damper system is given by:$$\mathrm{m\frac{d^{2}x}{dt^{2}}} + \mathrm{b\frac{dx}{dt}} + \mathrm{kx = F_{0}sin(\omega t)}$$where k is the spring constant.x(0) = 0 and x'(0) = 0.As we have the damping coefficient (b), we can calculate the damping ratio (ζ) and natural frequency (ωn) of the system.Damping ratio:$$\mathrm{\zeta = \frac{b}{2\sqrt{km}}}$$where k is the spring constant and m is the mass of the system.Natural frequency:$$\mathrm{\omega_{n} = \sqrt{\frac{k}{m}}}$$where k is the spring constant and m is the mass of the system.Resonant frequency:$$\mathrm{\omega_{d} = \sqrt{\omega_{n}^{2}-\zeta^{2}\omega_{n}^{2}}}$$At resonance, the amplitude of the system will be maximum when forced by a sinusoidal force of frequency equal to the resonant frequency.Resonant frequency:$$\mathrm{\omega_{d} = \sqrt{\omega_{n}^{2}-\zeta^{2}\omega_{n}^{2}}}$$$$\mathrm{\omega_{d} = \sqrt{(6.57)^{2}-(-2.88)^{2}} = 6.98 rad/s}$$Hence, the frequency of applied force at which resonance will occur is 6.98 rad/s.

The frequency of the applied force at which resonance will occur is ω = 2√5 rad/s.

To determine the frequency of the applied force at which resonance will occur, resonance happens when the frequency of the applied force matches the natural frequency of the system. The natural frequency can be determined using the formula:

ωn = √(K / M),

where ωn is the natural frequency, K is the spring constant, and M is the mass of the system.

Substituting the given values of K = 400 N/m and M = 20 kg into the equation, we can calculate the natural frequency ωn.

ωn = √(400 N/m / 20 kg) = √(20 rad/s²) = 2√5 rad/s.

Therefore, the frequency of the applied force at which resonance will occur is ω = 2√5 rad/s.

The correct question is given as,

M= 20kg

Fo = 90 N

ω = 6 rad/s

K = 400 N/m

C = 125 Ns/m

Determine the frequency of applied force at which resonance will occur?

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Consider a car speeding up as it drives along a level road. what is an action-reaction pair (from newton’s third law)?

Answers

Newton's third law of motion states that for every action, there is an equal and opposite reaction. The action-reaction pair is used to describe the interaction between two objects. Therefore, when a car is speeding up as it drives along a level road, an action-reaction pair occurs.

An action-reaction pair is a pair of forces that are equal in strength and opposite in direction. When an object exerts a force on another object, the second object exerts an equal and opposite force on the first object. The action-reaction pair when a car speeds up as it drives along a level road can be explained as follows: Action force: The car exerts a force on the road in the forward direction. This is the action force. Reaction force: The road exerts a force on the car in the backward direction. This is the reaction force.

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Can anyone please help me put

Can anyone please help me put

Answers

Answer:

Bulb a

Explanation:

Need all the answers to (Lab report: phase change) asap

Need all the answers to (Lab report: phase change) asap

Answers

Explanation: It's really blurry and sideways

By measuring the length of the skid marks, an accident investigator determines that the distance a car travelled between the brakes being applied and stopping was 22 m.

The investigator used a sled to determine the friction. The investigator then calculated that the car decelerated at 7.2 m / s2.
Calculate the speed of the car just before the brakes were applied.

Please help, i’m so confused!!

Answers

The initial velocity of the car just before the brakes were applied is 17.80 m/s

Given parameters

From the question given above, the following data were obtained:

Distance (s) = 22 mDeceleration (a) = -7.2 m/s²Final velocity (v) = 0 m/sInitial velocity (u) = ?

How to determine the initial velocity of the car

The initial velocity of the car can be obtained as illustrated below:

v² = u² + 2as

0² = u² + (2 × -7.2 × 22)

0 = u² + (-316.8)

0 = u² - 316.8

Collect like terms

u² = 0 + 316.8

u² = 316.8

Take the square root of both side

u = √316.8

u = 17.80 m/s

Thus, we can conclude that the initial velocity of the car before the brakes were applied is 17.80 m/s

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What is the wavelength of a sound wave that has a speed of 680 m/s and a frequency of 220 Hz?
7 What is the wavelength of a sound wave that has a speed of 960 m/s and a frequency of 875 Hz?
What is the wavelength of a sound wave that has a speed of 125 m/s and a frequency of 800 Hz?
9 What is the wave speed of a wave that has a frequency of 25 Hz and a wavelength of 4.0 m?
10 What is the wave speed of a wave that has a frequency of 400 Hz and a wavelength of 0.30 m?

Answers

The shift or change in a wave's frequency or wavelength brought on by the proximity of an emitting or reflecting sound source to the receiver. Therefore, the wavelength of the sound wave is 1.10 meters.

What does the Doppler effect look like in practise?

You may have noticed that the pitch of a fast-moving siren suddenly decreases as it passes past you. Because the pitch is higher, the siren initially appears to be approaching you. The siren's pitch drops as it moves away from you after passing you. The Doppler effect is being displayed here.

wavelength = speed / frequency

where wavelength is in units of meters, speed is in units of meters per second (m/s), and frequency is in units of Hertz (Hz).

For a sound wave with a speed of 680 m/s and a frequency of 220 Hz:

wavelength = speed / frequency

wavelength = 680 m/s / 220 Hz

wavelength = 3.09 m

Therefore, the wavelength of the sound wave is 3.09 meters.

For a sound wave with a speed of 960 m/s and a frequency of 875 Hz:

wavelength = speed / frequency

wavelength = 960 m/s / 875 Hz

wavelength = 1.10 m

Therefore, the wavelength of the sound wave is 1.10 meters.

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The voltage between two points in a circuit is 4.2 V. If the resistance between
the points is 72, what is the current, according to Ohm's law?
A. 0.058 A
B. 76.2 A
O C. 17.1 A
O D. 302 A

Answers

Answer:

I = 0.058 [A]

Explanation:

Ohm's law is defined as voltage equal to the product of current by resistance.

\(V =I*R\)

where:

V = voltage = 4.2 [V]

I = current [amp]

R = resistivity = 72 [ohm]

Now clearing I, we have:

\(I = V/R\\I = 4.2/72\\I = 0.058 [A]\)

Answer:

A. 0.058 A

Explanation:

Joey drives along the 110 South freeway and notices a mile marker that reads 260 miles. He drives until he reaches the 150 mile marker. He realizes that he went too far, so gets on the 110 North to the 175 mile marker. What is Joey’s displacement from the 260 mile marker?

Answers

Answer:

85 miles .

Explanation:

Displacement along the 110 South freeway = 260 - 150  =  110 miles

Displacement along the 110 North freeway = 150 - 175   = - 25 miles

Net displacement = 110 - 25 = 85 miles

So Joey's displacement from the 260 mile marker is 85 miles .

A trapezoid is a 4-sided polygon which has 2
parallel sides.
Write an expression that represents the
perimeter of a trapezoid whose sides have
length a, b, c and d.

Use this expression to find the perimeter of
the trapezoid when a= 12. b
16. c = 16.
and d = 20.​

Answers

796
I’m pretty sure that’s it

What type of Circuit do they use in a home? WHY ? (Simple circuit / Parallel circuit / Series circuit)

What type of Circuit do they use in a home? WHY ? (Simple circuit / Parallel circuit / Series circuit)

Answers

Answer:

Parallel Circuits

Explanation:

Loads can be operated on their own. If you had a series circuit, adding another light would dim the rest!

A compound microscope that has an eyepiece lens with a magnification of 10X and an objective lens with a magnification of 25X will have a total magnification of __________ times.

Answers

Answer:

35

Explanation:

total magnification = eyepiece lens x objective lens

TM = 10X x 25X

TM = 250X

Calculate the total charge Q of a thin plate with the charge density distribution p(x,y) = x²y mC/m². The plate shape is restricted by the lines: y=2-x² and y=2x-1, where x and y are measured in metres. a) Sketch the plate shape. [10 marks] [4 marks] b) Present the total charge through the double integral. c) Reduce the double integral to the repeated integrals and show limits of integration. [6 marks] d) Calculate the integral and present your answer with five significant figures. [20 marks]

Answers

a) Sketch the plate shape: we get a shape that resembles a trapezoid.

The plate shape is determined by the lines y = 2 - x² and y = 2x - 1. To sketch the plate shape, we can plot these two lines and shade the region in between them. The intersection points of the lines are found by solving the equations simultaneously:

2 - x² = 2x - 1

Simplifying, we get:

x² + 2x - 3 = 0

Factoring, we have:

(x - 1)(x + 3) = 0

So, x = 1 and x = -3. Plugging these values into the equations of the lines, we find the corresponding y-values:

For x = 1:

y = 2 - (1)² = 1

For x = -3:

y = 2(-3) - 1 = -7

Plotting these points and connecting them with the lines, we get a shape that resembles a trapezoid.

b) Total charge through the double integral:

To find the total charge Q, we need to integrate the charge density p(x, y) over the entire plate. We can express this as a double integral:

Q = ∬ p(x, y) dA

c) Reducing the double integral to repeated integrals: The limits of integration for x are the values of x that define the boundaries of the plate shape, which are -3 to 1.

Since the plate shape is described by the lines y = 2 - x² and y = 2x - 1, we can rewrite the double integral as a repeated integral by integrating with respect to x and y separately:

Q = ∫∫ p(x, y) dy dx

The limits of integration for y are from the lower curve y = 2 - x² to the upper curve y = 2x - 1. The limits of integration for x are the values of x that define the boundaries of the plate shape, which are -3 to 1.

d) Calculating the integral: The total charge Q of the thin plate is approximately 12.4 mC.

Now, we can evaluate the double integral to find the total charge Q:

Q = ∫(-3 to 1) ∫(2 - x² to 2x - 1) x²y dy dx

Performing the inner integral with respect to y first, we get:

Q = ∫(-3 to 1) [x²(y²/2 - y)] from 2 - x² to 2x - 1 dx

Simplifying the inner integral, we have:

Q = ∫(-3 to 1) [(x²/2)(2 - x²) - x²(2x - 1)] dx

Expanding and simplifying further, we get:

Q = ∫(-3 to 1) (x² - x⁴/2 - 4x³ + 2x²) dx

Integrating term by term, we have:

Q = [x³/3 - x⁵/10 - x⁴ + 2x³/3] from -3 to 1

Evaluating the integral at the limits, we get:

Q ≈ 12.4 mC (rounded to five significant figures)

Therefore, the total charge Q of the thin plate is approximately 12.4 mC.

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9. A 2.5 N book falls from a bookshelf 4.0 m above the floor. How much work does gravity do on the book as it falls 4.0m?

Answers

W=FS
2.5*4=10
The answer is 10 Joules of force

Which action might lead scientists to develop new explanations about the
universe?
A. Making a guess about how many stars are in the Milky Way galaxy
B. Wondering whether other planets can support life and whether
that life would be similar to life on Earth
C. Discovering new planets orbiting a star by detecting a wobble in
the star's motion
D. Being familiar with different types of galaxies that exist in the
universe

Answers

Answer:B

Explanation:

Being familiar with the different types of galaxies that exist in the universe is the action which might lead the scientists to develop new explanations about the universe. Thus, the correct option is D.

How did the universe developed?

Many religious person, including many great scientists, hold that the God has created the universe and the various processes which are the driving physical and biological evolution and that these processes then resulted in the creation of the different galaxies, the solar system, and life on the planet Earth.

The Big Bang theory states that the universe came into being from a single, unimaginably hot and dense point more than about 13 billion years ago. This did not occur in an already existing space. Rather, it initiated from the expansion and cooling of space itself.

Therefore, the correct option is D.

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