A car traveling at 60 mph has how much more energy than a car going at 15 mph?
KE increases by a factor of____.

1. 1/16
2. 4
3. 1/4
4. 16

Answer:

The correct answer is

right; fall

Explanation:

When there are changes in the costs of the factors of production, it can result in the supply curve shifting to the right or to the left. That means the quantity supplied at the given price has either increased or decreased, hence the supply curve shows the relationship between quantity supplied at a given price

Factors that can cause a shift in supply curve include, prices of imput materials, increased competition, technological advancement, social or natural factors and general expectations.

When the supply curve shifts right, that means increased in supply or more units are available at a given price, hence prices fall. However if the supply curve shifts left that means less goods are available at a given market price, hence prices of the commodities will rise

Answer:

[tex]\displaystyle Perpendicular[/tex]

Explanation:

Longitudinal waves are parallel to the direction of the motion of the disturbance, while transverse waves are perpendicular to the direction of the motion of the disturbance.

I am joyous to assist you anytime.

In a transverse wave, the motion of the disturbance is perpendicular to the wave motion.

In conclusion, in a transverse wave, the motion of the disturbance is perpendicular to the wave motion.

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Final answer:

The force exerted by the track on the cart at the top of the loop is 220 N, calculated by the difference between the centripetal force required for circular motion and the gravitational force acting on the cart.

Explanation:

The student is asking for the force exerted by the track on a 100 kg cart at the top of a vertical loop, where the cart has a speed of 6 m/s and the loop's radius is 3 m. At the top of the loop, the cart's weight and the normal force from the track provide the centripetal force necessary to keep the cart moving in a circle. To calculate the total force exerted by the track on the cart, we need to account for both the cart's weight (gravitational force) and the centripetal force required to keep it in circular motion.

The gravitational force (Fg) can be calculated using the equation Fg = m × g, where m is the mass of the cart and g is the acceleration due to gravity. The centripetal force (Fc) required for circular motion at the top of the loop can be calculated with the formula Fc = m × v² / r, where v is the speed of the cart and r is the radius of the loop.

The force exerted by the track (Ft) is then the difference between the centripetal force and the gravitational force because, at the top of the loop, the track has to push down on the cart to provide the inward centripetal force while also supporting the cart's weight.

Let's calculate each force:
Gravitational Force (Fg): Fg = 100 kg × 9.8 m/s² = 980 N
Centripetal Force (Fc): Fc = (100 kg) × (6 m/s)² / (3 m) = 1200 N

Now, we calculate the total force exerted by the track at the top of the loop:

Total Force by Track (Ft):
Ft = Fc - Fg
Ft = 1200 N - 980 N = 220 N

Therefore, the force exerted by the track on the cart at the top of the loop is 220 N.

Answer:

A. moral development progressed through distinct, qualitatively different stages.

Explanation:

Kolhberg’s theory of moral development states that we progress through three levels of moral thinking that build on our cognitive development.

Answer:

The period is (0.0017×wavelength) seconds

Explanation:

Speed = frequency × wavelength

Frequency = 1/period

Therefore, speed = 1/period × wavelength

Speed = wavelength/period

Period = wavelength/speed = wavelength/600 = (0.0017×wavelength) seconds

Answer:

A slow down and stop

Explanation:

When there is no force acting on something it automatically begins to slow down and then stops.Essentially, Aristotle's perspective of motion is that "it requires a force to move an object in an unnatural" way— or, plainly, that "movement involves strength." Indeed, if you propel a book, it keeps moving. Once you stop trying to push, it comes to a stop.

Increase the thickness of the wire would decrease the resistance in a wire

Explanation:

Thicker wires have a larger cross-section that increases the surface area with which electrons can flow unimpeded.  The thicker the wire, therefore, the lower the resistance.

Thin wires have very high resistance the reason the thin tungsten in a bulb glows because it is heated from the high resistance of many electrons trying to pass through a very small cross-section.

Answer:

Increase the thickness of the wire

Explanation:

I just took the test.

Answer:

Option B is correct.

Power = 360 W

Explanation:

Power = Work done/time

Work done = Force × distance moved through by the force

Power = Force × (distance moved through by the force/time)

(Distance moved through by the force/time) = velocity = 3 m/s

Power = Force × velocity

Force = ma

But the acceleration in this case is this acceleration + acceleration due to gravity because the force has to be overcoming the force of gravity to now move the object upward at 2 m/s²

a = (2 + g) (assume acceleration due to gravity = 10 m/s²

a = 2 + 10 = 12 m/s²

F = ma = 10 × 12 = 120 N

Power = F × v = 120 × 3 = 360 W

The student who has the correct analysis of the forces and motion of the backpack understands Newton's laws of motion and can apply them to the situation.

The student who has the correct analysis of the forces and motion of the backpack would be the one who understands the principles of Newton's laws of motion and can apply them to the situation.

Newton's first law of motion states that an object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same speed and in the same direction, unless acted upon by an external force. This law applies to the backpack on the group's graph, as it initially moves forward and then stops.

Newton's second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The student who correctly recognizes the forces and acceleration involved in the motion of the backpack would have the correct analysis.

Answer:

Explanation:

a) We have equation of motion s = ut + 0.5 at²

        Initial velocity, u = ?

        Acceleration, a = -9.81 m/s²  

        Time, t = 0.193 s

        Displacement, s = 0.587 m      

     Substituting

                      s = ut + 0.5 at²

                      0.587 = u x 0.193 + 0.5 x -9.81x 0.193²

                      u = 3.99 m/s

    Initial speed as it leaves the ground is 3.99 m/s

b) We have equation of motion v = u + at

     Initial velocity, u = 3.99 m/s

     Final velocity, v = ?

     Time, t = 0.193 s

     Acceleration, a = -9.81 m/s²

   Substituting

                      v = u + at  

                      v = 3.99 + -9.81 x 0.193

                      v = 2.10 m/s

    Speed at the height of 0.587 m is 2.10 m/s

c) We have equation of motion v² = u² + 2as

Initial velocity, u = 3.99 m/s  

Acceleration, a = -9.81 m/s²  

Final velocity, v = 0 m/s  

Substituting  

v² = u² + 2as

0² = 3.99² + 2 x -9.81 x s

s = 0.81 m

Height reached is 0.81 m

Answer:

233.43 N

Explanation:

Force: This is the product of mass and acceleration of a body.

The formula for force is given as,

F = ma .............. Equation 1

Where F = Net force on the bobsled, m = mass of the bobsled, a = acceleration of the bobsled

We can look for a using the equation of motion

v² = u² + 2as.............. Equation 2

Where V = final velocity, u = initial velocity, a = acceleration, s = distance.

making a the subject of the equation,

a = (v²-u²)/2s................... Equation 3

Given: v = 7.1 m/s, u = 0 m/s ( from rest), s = 8.1 m.

Substitute into equation 3

a = (7.1²-0²)/(2×8.1)

a = 50.41/16.2

a = 3.1 m/s²

Also given: m = 75.3 kg

Substitute into equation 1

F = 75.3×3.1

F = 233.43 N

Hence the net force on the bobsled = 233.43 N

Answer:

Isaac Newton supported his particle explanation of light by stating that light causes electrons to be emitted from a metal surface.

Explanation: The emission of electrons from metal surfaces when light of sufficient frequency falls on them is known as photoelecteic effect. He discovered photoelectric effect in the 19th century.

His other theories of light like light travels in a straight line could not explain diffraction and reflection. It needed the support of light as a wave to proof it.

Answer:

light travels in a straight line, and it casts a shadow.

Explanation:

Answer:

True

Explanation:

Modern safer and cheaper nuclear reactors can not only meet the range of our long term energy demands, they can also fight global warming.

Modern techniques provide ways to reduce radioactive waste amount. "A closed fuel cycle may be switched on for new kinds of nuclear plants. Alternatively, the waste is chemically dissuaded to transform the reusable element into fuel. This implies that nuclear waste would not be buried.

Answer:

Truee

Explanation:

Answer:

1.15 Billion Years

Explanation:

If the average age of the universe is 13.772 billion years and that equals to a year in cosmic calendar, then the length of time in a month will be 13.772 / 12 = 1.14766666667 ~ 1.15 Billion Years

Answer: 31.85 m/s

Explanation:

Since we are talking about constant acceleration, we can use the following equation to find the speed [tex]s[/tex] of the rocket:

[tex]s=\frac{d}{t}[/tex]

Where:

[tex]d=86 m[/tex] is the distance the rocket has traveled

[tex]t=2.7 s[/tex] is the time in which the rocket has traveled the distance [tex]d[/tex]

Solving the equation with the given data:

[tex]s=\frac{86 m}{2.7 s}[/tex]

[tex]s=31.85 m/s[/tex] This is the rocket's speed

Answer : The value of [tex]K_a[/tex] of the weak acid is, [tex]3.36\times 10^{-3}[/tex]

Explanation : Given,

Initial concentration = 0.090 M

pH = 1.80

First we have to calculate the hydrogen ion concentration.

[tex]pH=-\log [H^+][/tex]

[tex]1.80=-\log [H^+][/tex]

[tex][H^+]=0.0158M[/tex]

Now we have to calculate the [tex]K_a[/tex] of the weak acid.

The dissociation reaction of weak acid is:

                         [tex]HA\rightleftharpoons H^++A^-[/tex]

Initial conc.   0.090     0       0

At eqm.       (0.090-x)  x       x

x = 0.0158 M

The expression for dissociation constant is:

[tex]K_a=\frac{(x)\times (x)}{(0.090-x)}[/tex]

Now put all the given values in this expression, we get:

[tex]K_a=\frac{(0.0158)\times (0.0158)}{(0.090-0.0158)}[/tex]

[tex]K_a=3.36\times 10^{-3}[/tex]

Thus, the value of [tex]K_a[/tex] of the weak acid is, [tex]3.36\times 10^{-3}[/tex]

The value of Ka of the weak acid with a concentration of 0.090 M and has a pH of 1.80 is 3.36 × 10-³.

To calculate the Ka of an acid, we have to calculate the hydrogen ion concentration of the acid using the following expression:

pH = -log {H+}

1.80 = -log {H+}

{H+} = 0.0158M

The dissociation equation is given as follows:

HA ⇌ H+ + A-

Ka = 0.0158²/(0.090 - 0.0158)

Ka = 2.49 × 10-⁴/7.42 × 10-²

Ka = 0.336 × 10-²

Ka = 3.36 × 10-³

Therefore, the value of Ka of the weak acid with a concentration of 0.090 M and has a pH of 1.80 is 3.36 × 10-³.

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The charge contained within the width of the ring on the plastic disk is approximately [tex]\(1.89 \times 10^6\)[/tex] elementary charges (e).

To find the charge contained within the width of the ring, we'll first calculate the total charge of the disk and then subtract the charge within the inner circle defined by the radius of 0.5 cm.

1. Calculate the total charge of the disk:

  Total charge, Q_total = Q = +(2.05 ✕ 10^6)e

2. Calculate the area of the entire disk:

  Area_disk = π * R^2

            = π * (1.80 cm)^2

            = 10.17 cm^2

3. Calculate the area of the inner circle:

  Area_inner_circle = π * (0.50 cm)^2

                    = π * 0.25 cm^2

                    = 0.785 cm^2

4. Calculate the area of the ring:

  Area_ring = Area_disk - Area_inner_circle

            = 10.17 cm^2 - 0.785 cm^2

            = 9.385 cm^2

5. Calculate the charge density of the disk:

  Charge density, σ = Q_total / Area_disk

                    = (2.05 ✕ 10^6)e / 10.17 cm^2

                    ≈ 201,183.43 e/cm^2

6. Calculate the charge within the width of the ring:

  Charge_ring = σ * Area_ring

              ≈ 201,183.43 e/cm^2 * 9.385 cm^2

              ≈ 1.89 ✕ 10^6 e

So, the charge contained within the width of the ring is approximately 1.89 ✕ 10^6 e.

Answer: 63% of the final charge

Explanation:

When a capacitor is connected to a battery/power supply, the capacitor charges following the exponential law:

[tex]Q(t)=Q_0 (1-e^{-\frac{t}{\tau}})[/tex]

where

[tex]Q_0[/tex] is the final charge of the capacitor, which is

[tex]Q_0 =CV_0[/tex]

where C is the capacitance and [tex]V_0[/tex] the potential difference of the battery

t is the time

[tex]\tau[/tex] is the time constant of the circuit

Re-writing the equation,

[tex]Q(t)=CV_0 (1-e^{-\frac{t}{\tau}})[/tex]

After a time equal to one time constant,

[tex]t=\tau[/tex]

Therefore the charge on the capacitor will be

[tex]Q(\tau)=CV_0 (1-e^{-\frac{\tau}{\tau}})=CV_0(1-e^{-1})=0.63CV_0[/tex]

Which means 63% of the final charge.

Final answer:

After one time constant, the charge on a capacitor in an RC circuit will be approximately 63% of its maximum value. The time constant τ is calculated as the product of resistance R and capacitance C. The formula for the charge at any time is Q(t) = Q_{max}(1 - e^{-t/RC}).

Explanation:

When a capacitor charges in an RC circuit, it goes through a process described by an exponential function. After a time period equal to one time constant, represented by the symbol τ and given by the formula τ = RC (where R is the resistance in ohms and C is the capacitance in farads), the charge on a capacitor will reach approximately 63% of its maximum charge. This characteristic time, the time constant, determines how quickly the capacitor charges up to its maximum value, which is limited by the supply voltage.

The charge Q on the capacitor can be determined by the equation Q(t) = Q_{max}(1 - e^{-t/RC}) where Q(t) is the charge at time t, Q_{max} is the maximum charge, R is the resistance, C is the capacitance, and e is the base of the natural logarithm. When t equals the time constant τ (t = RC), the charge on the capacitor is Q(τ) = Q_{max}(1 - e^{-1}) which is approximately 63% of Q_{max}.

Answer:

A.) the inverse of the square of the distance separating them

Explanation:

Coulombs law states that "the force of attraction between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them."

Mathematically, F = kq1q2/r²

Where q1 and q2 are the charges

r is the distance between the charges.

According to the law, the force between two charged objects is related to the inverse of the square of the distance separating them.

Coulomb’s Law states that the force between two charged objects is directly related to the inverse of the square of the distance between them.

According to Coulomb’s Law, the force between two charged objects is related to (a) the inverse of the square of the distance separating them. This law states that the force (F) between two charges is directly proportional to the product of their charges (q1 and q2) and inversely proportional to the square of the distance (r) between them. It's written as F = k * q1 * q2 / r^2, where k is Coulomb's constant. Thus, if the distance between the objects is doubled, the force between them is reduced to one fourth of its original value.

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Answer:

2.57 cm or 2.57×10⁻² m

Explanation:

From coulomb's law,

F = kqq'/r²................. Equation 1

Where F = force between the charges, q and q' = The first and second charge respectively, k = constant of proportionality, r = distance between the charges.

Making r the subject of equation 1

r = √(kqq'/F)................ Equation 2

Given: F = 0.67 N, q = 1.2 μC  = 1.2×10⁻⁶ C, q' = 4.1×10⁻⁶ C

Constant: k = 9×10⁹ Nm²/C².

Substitute into equation 2

r = √( 1.2×10⁻⁶×4.1×10⁻⁶×9×10⁹/0.67)

r = √(66.09×10⁻³)

r = √(6.609×10⁻⁴)

r = 2.57×10⁻² m

r = 2.57 cm or 2.57×10⁻² m

Hence the distance between the charge = 2.57 cm or 2.57×10⁻² m

Answer:

[tex]F = 4.05\times 10^{9}\ N[/tex]

Explanation:

Given:

Exerts pressure

[tex]p = 2.41\times 10^{7}\ Pa[/tex]

Piston has an area

[tex]A = 168\ cm^{2][/tex]

We need to find the force exerted by piston.

Solution:

Formula of pascals law's is given as:

[tex]p=\frac{F}{A}[/tex]

Where:

F = Force applied

P = Pressure exerted

A = Cross sectional area

Substitute force and area in pascal's formula.

[tex]2.4\times 10^{7}=\frac{F}{168}[/tex]

Using cross multiplication.

[tex]F = (2.41\times 10^{7})\times 168[/tex]

[tex]F = 404.88\times 10^{7}[/tex]

[tex]F = 4.05\times 10^{9}\ N[/tex]

Therefore, the force exerted by piston [tex]F = 4.05\times 10^{9}\ N[/tex]

The piston exerts a force of 404,880 Newtons when the motor exerts a pressure of 2.41 x 10⁷ Pa on a fluid chamber with a piston area of 168 cm².

To determine the force exerted by the piston, we use the formula:

Force (F) = Pressure (P) × Area (A)

Given:

First, convert the area from cm² to m²:

1 m² = 10,000 cm²

A = 168 cm² / 10,000 = 0.0168 m²

Now, calculate the force:

F = 2.41 × 10⁷ Pa × 0.0168 m²

F = 404,880 N

Thus, the piston exerts a force of 404,880 Newtons.

Answer:

P = 192.6 Watt

Explanation:

given,

Force of Pull, F = 63 N

Weight of the sled, W = 308 N

Angle made with ground, θ = 33°

Movement of sled,s= 11.3 m

time, t = 3.1 s

Power of Wayne = ?

Work done = F .s cos θ

W = 63 x 11.3 x cos 33°

W = 597 J

We know.

[tex]Power = \dfrac{Work\ done}{time}[/tex]

[tex]P= \dfrac{597}{3.1}[/tex]

  P = 192.6 Watt

The Wayne's Power is equal to P = 192.6 Watt

Answer:

-14892.93 J

Explanation:

given,

Pressure, P = 54700 Pa

heat removed, Q = 16800 J

radius, r = 0.272 m

distance, d = 0.150 m

internal energy ΔU of the system = ?

We know,

Force = Pressure x area

F = P x A

F = 54700 x π x r²

F = 54700 x π x 0.272²

F = 12713.79 N

Work done = F.d

W = 12713.79 x 0.15

W = 1907.07 J

Change in internal energy ΔU is

ΔU = W + Q

     = 1907.07 + (-16800)

     = -14892.93 J

Hence, the change in internal energy is equal to -14892.93 J

Answer:

This process involves the motion of dislocations and is termed slip (or glide in some textbooks)

Explanation:

Plastic deformation of metals (and other crystalline materials) usually occurs by slip, which is the sliding of planes of atoms over one another by dislocation movements.

On a microscopic scale, stress causes planes of crystalline objects to leave their original position and slide over other planes into new positions, these microscopic movements manifest as a slip on a macroscopic scale. And the planes do not return back to their original position after the removal of the dislocation-causing stress.

Answer:

For Pentium 4 Prescott:

% of Static Power = 10

For core i5 Ivy Bridge:

% of Static Power = 43

Given Information:

Static Power of P4 = 10 W

Dynamic Power of P4 = 90 W

Static Power of i5 = 30 W

Dynamic Power of i5 = 40 W

Required Information:

% of static power w.r.t total power dissipation = ?

Explanation:

For Pentium 4 Prescott:

% of static power = static power/total power * 100

% of static power = 10/(10 + 90) * 100

% of static power = 10/(100) * 100

% of static power = 10

For core i5 Ivy Bridge:

% of static power = static power/total power * 100

% of static power = 30/(30 + 40) * 100

% of static power = 30/(70) * 100

% of static power = 43 (rounded to nearest whole integer)

Final answer:

The static power comprises 10% of the total dissipated power for the Pentium 4 Prescott processor.

Explanation:

To calculate the percentage of the total dissipated power that is comprised of static power for the Pentium 4 Prescott, we need to add the static power to the dynamic power to get the total power and then find what percentage the static power is of the total power.

Total power for Pentium 4 Prescott = Static power + Dynamic power
Total power for Pentium 4 Prescott = 10 W + 90 W
Total power for Pentium 4 Prescott = 100 W

Percentage of static power = (Static power / Total power)  imes 100
Percentage of static power = (10 W / 100 W)  imes 100
Percentage of static power = 10%

So, the static power comprises 10% of the total dissipated power for the Pentium 4 Prescott.

Answer:

3.6 * 10^-13 C

Explanation:

The net charge of the sphere will be the sum of the total electron charge and total proton charge. Mathematically,

Q = Qe + Qp

TOTAL ELECTRON CHARGE:

An electron has an electronic charge of -1.6022 * 10^-19C.

Hence, the charge of 5.87 * 10^6 electrons will be:

Qe = - 1.6022 * 10^-19 * 5.87 * 10^6

Qe = - 9.405 * 10^-13 C

TOTAL PROTON CHARGE:

A proton has an electronic charge of 1.6022 * 10^-19. Hence, 8.11 * 10^6 protons will have:

Qp = 1.6022 * 10^-19 * 8.11 * 10^6

Qp = 1.3 * 10^-12 C

=> Q = (-9.405 * 10^-13) + (1.3 * 10^-12)

Q = 3.6 * 10^-13 C

Answer:

3.58*10⁻¹³ C

Explanation:

When we have a net charge on a sphere, this means that there must be a difference in the number of electrons and protons on the sphere, otherwise, the sphere would be electrically neutral.

In this case, we can find this difference as follows:

Np -Ne = 8.11*10⁶ protons - 5.87*10⁶ electrons = 2.24*10⁶ protons.

The total charge carried by all these protons is just the product of the charge of only one proton (which is equal to the elementary charge e) times the number of excess protons, as follows:

Qnet = 2.24*10⁶ protons * 1.6*10⁻19 C/proton = 3.58*10⁻¹³ C

Answer:

A. As the ropes are horizontal the child has travelled 2m of vertical displacement from his lowest position.

Gpe @ A=mgh=40*9.81*2=784.8J

B. At 30degree vertical angle the vertical displacement from lowest position is given by

2-2cos(30)=2-1.73=0.27m

Gpe @B= 40*9.81*0.27=106 J

C: at the bottom of circular arc it's Gpe is zero relative to lowest position as bottom of arc itself is lowest position.

(a) The gravitational potential energy when the ropes are horizontal is 784.8 J.

(b)   The gravitational potential energy when the ropes make a 30 degree angle with vertical is 106 J.

(c)  At the bottom of circular arc the gravitational potential energy is zero.

Given data:

The mass of child is, m = 40.0 kg.

The length of ropes is, L = 2 m.

The energy possessed by any object under the influence of gravity and by the virtue of position of object is known as gravitational potential energy.

(a)

As the ropes are horizontal the child has travelled 2m of vertical displacement from his lowest position. Then the gravitational potential energy is,

[tex]PE = mgL\\\\PE = 40 \times 9.8 \times 2\\\\PE=784.8 \;\rm J[/tex]

Thus, the gravitational potential energy when the ropes are horizontal is 784.8 J.

(b)

At 30 degree vertical angle the vertical displacement from lowest position, the gravitational potential energy is given by,

[tex]PE = mgL(1-cos30^{\circ})\\\\PE = 40 \times 9.8 \times 2 \times (1-cos30^{\circ})\\\\PE = 106 \;\rm J[/tex]

Thus, the gravitational potential energy when the ropes make a 30 degree angle with vertical is 106 J.

(c)

At the bottom of circular arc it's gravitational potential energy is zero relative to lowest position as bottom of arc itself is lowest position.

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Answer:

4, A proton in a particle accelerator moves faster and faster.

Explanation:

A particle accelerator according to Wikipedia.com is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for basic research in particle physics.

As the proton moves around, it collides with other proton and the wall of the accelerator thereby undergoing net interaction.

Answer:

a.3.86 Km

b.Direction=233.3 degree

Explanation:

We are given that

Person wale in North direction,DE=2.3 Km

Person walk in West direction==BD=2.5 Km

Person walk in South direction,EC=5.4 Km

a.We have to find the distance  between initial and final position.

Vector AB=-2.5 Km

BD=-EA=-2.3 Km

CA=y=-(EC-EA)=-(5.4-2.3))=-3.1 m

AB=-DE=x=-2.5 Km

Magnitude of resultant vector=[tex]\sqrt{AB^2+CA^2}=\sqrt{(-2.5)^2+(-3.1)^2}[/tex]

Magnitude of resultant vector=3.86 Km

Hence, the distance between initial and final position=3.86 Km

b.Direction=[tex]\theta=tan^{-1}(\frac{y}{x})[/tex]

Direction=[tex]\theta=tan^{-1}(\frac{-3.1}{-2.3})\approx 53.3^{\circ}[/tex] S of W

x-coordinate and y-coordinate are negative therefore, the angle lies in third quadrant.

When we measured from East then

[tex]\theta=53.3+180=233.3^{\circ}[/tex]

Therefore, the direction=233.3 degree

Answer:

16 times

Explanation:

Generally, the light-gathering power of the mirror of a telescope is dependent on the area of the mirror. The area of the mirror is (π*d^2)/4. The variable 'd' is the diameter of the mirror. Therefore, if the diameter of A is four times the diameter of B, the light-gathering power of A will be (π*4^2)/4 while that of B will be (π*1^2)/4. This shows that A has 16 times that of B.

A mirror with four times the diameter of another mirror collects 16 times more light than the smaller mirror in the same amount of time.

The light-gathering power of a telescope is directly proportional to the area of its primary mirror. The area A of a circular mirror is given by the formula [tex]\( A = \pi \left(\frac{D}{2}\right)^2 \)[/tex], where D is the diameter of the mirror.

If one mirror has four times the diameter of another, we can denote the diameters as D for the smaller mirror and [tex]\( 4D \)[/tex] for the larger mirror.

The area of the smaller mirror is:

[tex]\[ A_{\text{small}} = \pi \left(\frac{D}{2}\right)^2 = \pi \frac{D^2}{4} \][/tex]

The area of the larger mirror is:

[tex]\[ A_{\text{large}} = \pi \left(\frac{4D}{2}\right)^2 = \pi \left(2D\right)^2 = \pi 4D^2 \][/tex]

To find out how many times more light the larger mirror collects compared to the smaller mirror, we divide the area of the larger mirror by the area of the smaller mirror:

[tex]\[ \frac{A_{\text{large}}}{A_{\text{small}}} = \frac{\pi 4D^2}{\pi \frac{D^2}{4}} = \frac{4D^2}{\frac{D^2}{4}} = 4 \times 4 = 16 \][/tex]