A clock pendulum oscillates at a frequency of 2.5 Hz . At t=0, it is released from rest starting at an angle of 13 ∘ to the vertical.
Part A Ignoring friction, what will be the position (angle in radians) of the pendulum at t = 0.25 s ? Express your answer using two significant figures.
Part B Ignoring friction, what will be the position (angle in radians) of the pendulum at t = 2.00 s ? Express your answer using two significant figures.
Part C Ignoring friction, what will be the position (angle in radians) of the pendulum at t = 540 s ? Express your answer using two significant figures.
Answers
The position (angle in radians) of the pendulum is
A. At t = 0.25 s,0.20 rad.
B. At t = 2.00 s, -0.23 rad.
C. At t = 540 s,0.23 rad.
Step 1. The motion of the pendulum is described by the equation
θ(t) = θ0 cos(ωt)
Where θ0 is the amplitude (initial angle), ω is the angular frequency (ω = 2πf), and t is time.
Given: f = 2.5 Hz, θ0 = 13° = 0.23 rad
Step 2. Part A
At t = 0.25 s,
θ = θ0 cos(ωt)
θ = 0.23 cos(2π2.50.25)
θ = 0.20 rad.
Step 3. Part B
At t = 2.00 s,
θ = θ0 cos(ωt)
θ = 0.23 cos(2π2.52.00)
θ = -0.23 rad
(Note: the negative sign indicates that the pendulum has swung to the left of its equilibrium position)
Step 4. Part C
At t = 540 s,
θ = θ0 cos(ωt)
θ = 0.23 cos(2π2.5540)
θ = 0.23 rad
(Note: after a long time, the pendulum settles into a steady oscillation with constant amplitude and frequency)
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Related Questions
Question 3 of 5What could you do to increase the electric force between two chargedparticles by a factor of 16?A. Reduce one particle's charge by a factor of 16.B. Reduce one particle's charge by a factor of 4.C. Increase one particle's charge by a factor of 16.O D. Increase one particle's charge by a factor of 4.
Answers
Explanation
the force between 2 charges is given by the formula:
\(\begin{gathered} F=\frac{Kq_1q_2}{r^2} \\ where \\ q_i\text{ is the charge on object i} \\ r\text{ is the distance between the charges} \\ K=Coulomb\text{ constant=8.9875*10}^9N*\frac{m^2}{c^2} \end{gathered}\)then,we can check every option and compare the result to the original, hence
Step 1
a)option A
reduce one particle's charge by a factor of 16
so
\(q^{\prime}_1=\frac{q_1}{16}\)now, replace
\(\begin{gathered} F=\frac{Kq_1q_2}{r^2} \\ F_A=\frac{K\frac{q_1}{16}q_2}{r^2}=\frac{1}{16}\frac{Kq_1q_2}{r^2} \\ F_A=\frac{1}{16}(original) \\ F_A=\frac{1}{16}\frac{Kq_1q_2}{r^2} \end{gathered}\)therefore, option A works
Step 2
A cyclist is riding his bike up a mountain trail. When he starts up the trail, he is going 8 m/s. As the trail gets steeper, he slows to 3 m/s in 1 minute. What is the cyclist’s acceleration? –0.08 m/s2 0.08 m/s2 –12 m/s2 12 m/s2
Answers
Answer:
-0.08m/s²
Explanation:
The ball and the shoulder area ABOVE the armpit of an attacker make contact and the ball goes into goal. What is the decision by the referee?
Answers
The referee should award a goal to the attacker since the part of the body in contact with the ball is not hand and it cannot be considered as handball.
Goal rules for soccerThere are several rules considered during a game of soccer before the referee will award goal.
Some of the rules include;The ball must cross the goal lineThe attacker must be on sideThe attacker must play the ball with any part of the body EXCEPT the hand.The shoulder area ABOVE the armpit is not part of the hand, thus the referee should award a goal to the attacker since the part of the body in contact with the ball is not hand and it cannot be considered as handball.
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what is the frequency of an x-ray with wave-length 0.12 nm ?
Answers
The frequency of the x-ray is 2.5 × 10¹⁸ Hz. The result is obtained from the formula for frequency.
The frequency of an electromagnetic waveThe x-ray is one of the electromagnetic waves. The equation that relates frequency and wavelength of the x-ray is
c = λ × f
Where c = speed of light, λ = wavelength, and f = frequency.
First, we need to convert the given wavelength from nanometers (nm) to meters (m):
0.12 nm = 0.12 × 10⁻⁹ m
Now, we can rearrange the equation to solve for frequency:
f = c / λ
Next, plug in the values for the speed of light (c = 3 × 10⁸ m/s) and the wavelength (λ = 0.12 × 10⁻⁹ m):
f = (3 × 10⁸ m/s) / (0.12 × 10⁻⁹ m)
f ≈ 2.5 × 10¹⁸ Hz
So, the frequency of an x-ray with a wavelength of 0.12 nm is approximately 2.5 × 10¹⁸ Hz.
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Consider an aircraft powered by a turbojet engine that has a pressure ratio of 12. The aircraft is stationary on the ground, held in position by its brakes. The ambient air is at 300 K and 95 kPa and enters the engine at a rate of 10 kg/s. The jet fuel has a heating value of 42,700 kJ/kg, and it is burned completely at a rate of 0. 2 kg/s. Neglecting the effect of the diffuser and disregarding the slight increase in mass at the engine exit as well as the inefficiencies of engine components, determine the force that must be applied on the brakes to hold the aircraft stationary, in kN
Answers
The force that must be applied on the brakes to hold the aircraft stationary is 84.51 kN (approx).
The force that must be applied on the brakes to hold the aircraft stationary is 84.51 kN.
Step 1:We can determine the exhaust gas temperature using the formulaT4 = T3 + (qin / (cp * m_dot)).Where,T3 = 1191.7 K (Given)qin = heating value of fuel × mass flow rate of fuel = 42,700 × 0.2 = 8,540 kJ/kgcp = specific heat at constant pressure = 1.005 kJ/kg.Km_dot = mass flow rate of air + mass flow rate of fuel = 10 + 0.2 = 10.2 kg/s
Putting all the values,T4 = 1191.7 + (8,540 / (1.005 * 10.2)) = 1191.7 + 84.85 = 1276.55 K
Step 2:The static thrust produced by the engine can be found out by the formula:
F = m_dot * (V2 - V1) + A2 * (P2 - P1)
Where,m_dot = mass flow rate of airA2 = area of the exhaust nozzleV1 = velocity of air at the engine inletV2 = velocity of air at the engine exitP1 = pressure of air at the engine inletP2 = pressure of air at the engine exitWe can use the simplified form of the above formula, which neglects the increase in the kinetic energy of the air in the combustor, the increase in the potential energy of the air from the inlet to the exhaust, and the change in the kinetic energy of the air from the inlet to the exhaust.F = m_dot * (Ve - V1) + Ae * (Pe - P1)
Where,Ve = velocity of air at the engine exitAe = area of the exhaust nozzlePe = pressure of air at the engine exitWe can calculate the value of Ve using the formula:Ve = sqrt(2 * cp * T4 * (1 - (P2 / P1)^((γ - 1) / γ))))Where,γ = ratio of specific heats = 1.4Putting all the values,Ve = sqrt(2 * 1.005 * 1276.55 * (1 - (1 / 12)^0.4))) = 724.23 m/sNow, we can calculate the static thrust,F = m_dot * (Ve - V1) + Ae * (Pe - P1)= 10.2 * (724.23 - 0) + (0.785 * (0.25)^2 * (12 * 10^3 - 95) * 10^3)= 74,044.47 N
Step 3:The force that must be applied on the brakes to hold the aircraft stationary is equal and opposite to the static thrust, that is, 74,044.47 N.
The value in kN will be: 74.04447/1000 = 74.04 kN.
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In 2 - 3 sentences, explain the difference in Kinetic energy and Potential energy.
Answers
The main difference between Kinetic energy and Potential energy is that kinetic energy refers to movement while potential energy refers to storage.
What are Kinetic energy and Potential energy?Kinetic energy is a type of energy in motion or energy in movement such as a turbine (mechanical energy), while Potential energy refers to the energy that is stored to be used when required (e.g. chemical bonds of foods).
Therefore, with this data, we can see that Kinetic energy is used as movement, while Potential energy is stored to be used in the future.
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Coulomb’s law is stated mathematically as
F
=
k
q
1
q
2
r
2
where F represents electrical force, k is the electrostatic constant, q represents strength of an electrical charge and r represents the distance between the centers of the charges.
• Explain the relationship between force and strength of electrical charge and the relationship between force and distance, based on this equation.
• Determine how doubling the strength of both charges at the same time affects the force between them.
Answers
Relationship between force and strength of electrical charge directly proportional whereas the relationship between force and distance is inversely proportional
Relationship between force, charges and distanceThe relationship between force and strength of electrical charge directly proportional to each other which means if one increases the other will automatically increases whereas the relationship between force and distance is inversely proportional which means that if one increases the other will decrease.
Doubling the strength of both charges at the same time affects the force between them because both charges are directly proportional to its force so we can conclude that relationship between force and strength of electrical charge directly proportional whereas the relationship between force and distance is inversely proportional.
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(Repost) I need help with this physics question. Thanks in advance! Answer ASAP.
Answers
(a) The gravitational force received by each 1 kg mass is 8.66 N.
(b) The magnitude of gravitational acceleration is 8.66 m/s².
(c) The orbital speed of the ISS is 7,663.6 m/s.
(d) The time take for the ISS to orbit round the Earth is 5,558.75 seconds which is equal to 1.54 hours.
Gravitational force received by each 1 kg massThe gravitational force received by each 1 kg mass is calculated as follows;
F = Gm₁m₂/r²
where;
m₁ is mass of Earthm₂ is mass of ISSr is the distance between the ISS and center of EarthF = (6.67 x 10⁻¹¹ x 5.97 x 10²⁴ x 1) / (6780,000)²
F = 8.66 N
Magnitude of gravitational accelerationmg = GMm/r²
g = GM/r²
where;
M is mass of Earthg = (6.67 x 10⁻¹¹ x 5.97 x 10²⁴ ) / (6780,000)²
g = 8.66 m/s²
Orbital Speed of the ISSv = √GM/r
v = √(6.67 x 10⁻¹¹ x 5.97 x 10²⁴ / 6780,000)
v = 7,663.6 m/s
Time of motion of the ISS round the EarthT = 2πr/v
T = (2π x 6780,000) / (7663.6)
T = 5,558.75 seconds
1 hour = 3600 seconds
= 5,558.75/3600
= 1.54 hours
Thus, the gravitational force received by each 1 kg mass is 8.66 N.
The magnitude of gravitational acceleration is 8.66 m/s².
The orbital speed of the ISS is 7,663.6 m/s.
The time take for the ISS to orbit round the Earth is 5,558.75 s = 1.54 hours.
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What is mean by the power of lens is 2 diopter?
Answers
The power of the lens is 2 diopter means that this lens has a focal length of 1/2 meter.
The focal length = 1 ÷ ( the number of diopters)
If one's glass prescription shows that the power of the lens is 2 diopter it means that the person is far-sighted. It means if a person needs to see something which is quite distant then his eyes need extra lens power to focus on it. Moreover, if there's a sign convention it means '+' indicates a convex lens and '-' concave lens. Also in-camera close up lens are also measured in terms of diopter.
I hope this answer helps.
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You are driving at 25 m/s with your cruise control on when you see a fallen tree in the road. It takes you 0.30 s to put on the brakes, your car decelerates at a rate of 5.0 m/s2, and the tree is 75 m ahead. Will you hit the tree? Choose the correct answer and reason below.
Answers
Explanation:
Given data
velocity v= 25m/s
The time it takes to put on brake t= 0.3s
the distance covered when the brake was put on is
v=s/t
s= v*t
s= 25*0.3s
s= 7.5m
hence the distance covered is 7.5m
Also the rate of decrease in aceleration is 5m/s^2
we can also calculate the distance covered at this rate
v^2=u^2+2as
25^2= 0+2*5*s
625=10s
divide both sides by 10
s=625/10
s= 62.5m
The total distance covered between putting on the brakes and decelareation is 7.5+62.5= 70m
Given that the tree is 75m ahead, the car would not hit the tree
why is fundamental unit called derived unit ?
Answers
Since fundamental units are elementary in nature and cannot be reduced further, so it can not be reduced further, so it cannot be expressed in other units
suppose two protostars form at the same time, one with a mass of 0.5 msun and the other with a mass of 15 msun . which of the following statements are true?
Answers
15 M sun star will end its main sequence life earlier. the 15M sun protostar will be more luminous.
How is a large protostar different from a small protostar?Outer space gas clouds are where stars first start to develop. These clouds' frigid temperatures and high densities (in comparison to other regions of space, but in comparison to a vacuum on Earth) enable gravity to overcome thermal pressure and initiate the gravitational collapse that would result in the formation of a star.A protostar resembles a star, but its core is still too cool for fusion to occur. The protostar's heating as it contracts is the only source of brightness. The light that protostars release is typically blocked by dust, making them challenging to study in the visible spectrum.To learn more about protostar, refer to
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You look at a rope coiled on a beach and are able to perceive it as a single strand because of the law of.
Answers
You are able to perceive it as a single strand because of the principle of : Good continuation
Principle of good continuationPrinciple of Good continuation states that objects with very smooth edges are easily seen as been continuous ( i.e as a single strand ) rather than objects or materials with rough edges.
The rope coiled on a beach has very smooth edges therefore can be seen as a continuous strand and this is possible due to the principle of good continuation.
Hence we can conclude that You are able to perceive it as a single strand because of the principle of : Good continuation.
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what is the equation for wave velocity
Answers
Answer:
v = λf
Explanation:
The equation for wave velocity is:
v = λf
where v is the wave velocity, λ (lambda) is the wavelength, and f is the frequency of the wave.
a body is thrown at an angle 30°to the horizontal with kinetic energy 40J.atbthe highest point of its motion the kinetic energy will be?
Answers
how is an unmagnetized piece of iron different from the same piece of iron when it is magnetized?
Answers
An unmagnetized piece of iron differs from a magnetized piece of iron primarily in their internal structure and magnetic properties. In an unmagnetized iron, the individual magnetic domains are randomly oriented, which means that their magnetic fields cancel each other out, resulting in no net magnetic field.
On the other hand, when the iron becomes magnetized, the magnetic domains align in the same direction, thereby creating a net magnetic field. This process is typically achieved through the application of an external magnetic field or by placing the iron piece in close proximity to a strong magnet. The alignment of the magnetic domains causes the magnetized iron to exhibit magnetic properties, such as attracting or repelling other magnetic materials and generating a magnetic field.
In terms of practical applications, magnetized iron can be used in a variety of devices, such as electromagnets, transformers, and magnetic storage media. Unmagnetized iron, while not possessing these magnetic properties, still retains its inherent characteristics, like strength and ductility, making it suitable for structural and engineering purposes.
In summary, the main difference between an unmagnetized and magnetized piece of iron lies in the orientation of their magnetic domains and the resulting presence or absence of a net magnetic field. This difference gives rise to distinct magnetic properties, which ultimately determine their specific applications in various industries.
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two parallel wires are 6.00 cm apart, and each carries a current of 1.00 a. (a) if the currents are in the same direction, find the force per unit length exerted on one of the wires by the other. are the wires attracted to or repelled by each other? (b) repeat the problem with the currents in opposite directions.
Answers
(a) Same-direction currents result in attraction. Force per unit length: 1.11 × 10⁻⁵ N/m.
(b) Opposite direction currents result in repulsion. Force per unit length: -1.11 × 10⁻⁵ N/m.
When two parallel wires carry currents in the same direction, they experience a force of attraction, while they experience a force of repulsion when the currents are in opposite directions. The force per unit length exerted on one of the wires by the other can be calculated using the formula F = μ₀I₁I₂L / (2πd), where F is the force per unit length, μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, L is the length of the wires, and d is the distance between the wires. For the given problem, the force per unit length exerted on one of the wires by the other is 1.11 × 10⁻⁵ N/m when the currents are in the same direction, indicating attraction, and -1.11 × 10⁻⁵ N/m when the currents are in opposite directions, indicating repulsion. Therefore, the wires either attract or repel each other based on the direction of the currents they carry.
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In a particular chemical reaction, the energy of the reactants is 30 kJ and the energy of the products is 5 kJ. The maximum energy of the system is 40 kJ.
Sketch a potential energy diagram for this reaction. Make sure to label the energy of the reactants, the energy of the products, the activation energy, and the enthalpy change for the reaction.
What is the activation energy for this reaction?
What is the enthalpy change for this reaction?
Is this reaction endothermic or exothermic? Explain your answer in two ways: first, using the energy values, and second, by referring to the shape of the graph.
The coating on the head of a match is highly flammable. When it burns, it releases a great deal of energy. However, before the match can burn, it must gain a small amount of energy from a spark. That spark is typically produced by striking (rubbing) the match head against a rough surface. Sketch and describe a potential energy diagram that represents the striking and burning of the match. Remember to label the diagram with the energy changes that occur. Your answer must include the potential energy diagram and a written description. (Note: you do not have to use actual energy values.)
Answers
An energy of 25kJ is given out hence the reaction is exothermic.
What is a reaction?Chemical reaction is said to occur when reactants combine leading to the formation of products. In a plot of the energy against the reaction profile, the topmost point of the graph is called the activation energy.
We know that a reaction is exothermic when reactants posses more energy than the products. Hence the enthalpy change for this reaction is;
ΔH = 5kJ - 30kJ = -25kJ thus the reaction is an exothermic reaction.
Now, the activation energy for the lighting of the match stick is high that is why a spark is required before the reaction gets underway. Heat is evolved in the process hence the reaction is exothermic. The energy of the reactants is greater than the energy of the products.
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a linebacker strikes a side referee with a force of 100 n (on accident of course). the linebacker is in contact with the referee for 0.10 s. what is the magnitude of the change in momentum of the referee?
Answers
The magnitude of the change in momentum of the referee will be 10 Ns.
What does a momentum unit change mean?Mass times velocity equals mass (M) times momentum (P) (v). However, there are alternative perspectives on momentum. The change in momentum (P) over the time period (t) equals force (F). And the impulse is equal to the shift in momentum (P) (J). The units for momentum and impulse are kg*m/s and N*s, respectively.
How to calculate change in momentum or Impulse?Given, Force = 100N
time = 0.1. s
Impulse, J = F.t
J = 100×0.1
J or Δp = 10 Ns
The magnitude of the change in momentum of the referee will be 10 Ns.
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Objects in motion have kinetic energy because of their___ and ___
Answers
Answer:
Mass and velocity
Explanation:
Velocity is the speed at which the object is moving, so imagine getting hit with a bike versus car, and mass is the difference between a feather and book. Both can contribute to total kinetic energy.
4 "Find shelter in a building or a metal-topped vehicle (not a convertible), and close the windows. " Based on this safety tip, what can you conclude about the safety of cars during lightning? A All cars, regardless of type, will protect you from lightning B B Cars are less safe than buildings during lightning. The rubber tires of a car protect you from lightning. D a The metal roof of a car protects you from lightning.
Answers
Answer :- Option D
Explanation:-
Coz electric field inside a conductor is zero that's why we don't get electrocuted.
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An antibaryon composed of two antiup quarks and one antidown quark would have a charge of:.
Answers
A charge of -1e would be present in an antibaryon made up of two antiup quarks and one anti-down quark. Option 2 is correct.
What is the charge?When the matter is put in an electromagnetic field, it has an electric charge, which causes it to experience a force. A positive or negative electric charge can exist.
The complete question is
"An antibaryon composed of two antiup quarks and one anti down quark would have a charge of
(1) +1e
(2) −1e
(3) 0e
(4) −3e"
An antibaryon composed of two antiup quarks and one anti-down quark would have a charge of −1e.
Hence,option 2 is correct.
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If an object starts at rest, what is necessary to make it start moving?
Answers
It will need a force acceleration
Reflecting telescopes use mirrors to bring light to a focus. All large telescopes are reflectors because_____
Answers
All large telescopes are reflectors since they offer these benefits and are better suited for large telescopes.
Specifically: It is feasible to manufacture telescopes with wider apertures (larger diameter mirrors), which can collect more light and provide higher resolution images, because mirrors can be produced larger than lenses without sagging under their own weight.
Thin metal sheets can be applied to mirrors to improve reflectivity and decrease light loss.
Mirrors can be supported from the back, which increases their rigidity and solves the lens flexure issues that plague refracting telescopes.
Although producing huge lenses to exact specs is more expensive and complex than producing large mirrors, reflectors are typically less expensive to produce than refractors.
Hence, all large telescopes are reflectors since they offer these benefits and are better suited for large telescopes.
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shape of houses built in areas with hot climate and materials used
Answers
Answer:
Cement and sand will do
Explanation:
because without it you can't build
Answer:
Shape of houses built in areas with hot climate
Compact house forms, rather than sprawling, multi-wing designs work best in hot locales. Home designs with shaded porches, wrap around porches or plans with courtyards that create a shaded open area within the building are ideal.
How do you build a house in hot climates?
Place courtyards centrally to enhance cooling and ventilation. ...
Choose the right materials. ...
Use waterbodies and landscaping effectively. ...
Shade terraces and outdoor spaces. ...
Create sheltered, barrier-free spaces for air circulation. ...
Extend roofs to protect the walls and windows from direct heat.
Materials used building in areas with hot climate
The most common materials used for building walls in the warm-humid regions are stones, and earth-based bricks.
Concrete, bricks, and tile have a high thermal mass due to the large amount of heat energy needed to change their temperatures. This makes concrete an ideal building material for climates that are hot during the day but cooler at night.
1pt If an object is moving at a constant velocity, it means that....
A. no net force is acting on it
B. a constant force is acting on it in the direction of its motion
c. a constant force is acting on it in the opposite direction of its motion
D. several forces are acting on it, but one force is greater than the others
Answers
Answer:
thecool me
Explanation:
can someone please help me please I will mark brainlist and give you 5 stars pleasee
Answers
Answer:
If I did my calculations correctly then the answer is 11 but I may be weong
Answer:
it's 11
gkccgkxcgikvxxkg
Radio waves have wavelengths longer than 1 m. A: True B: False Blue light has a higher frequency than X-rays. A: True B: False Ultraviolet radiation causes common sunburn. A: True B: False A vertical automobile antenna is sensitive to electric fields polarized horizontally. A: True B: False Gamma rays travel in vacuum at the same speed as the visible light. A: True B: False X-rays can be produced in transitions involving inner electrons in an atom. A : True B : False The sun's radiation is most intense in the visible region. A: True B: False
Answers
The given statements majorly discusses about the various radiations as well as different wavelengths of the radiations. In the following statements, the statements 1,3,4,6,7 are true.
Radio waves have wavelengths longer than 1 m: This statement is true. Radio waves have wavelengths ranging from a few millimeters to hundreds of meters, and they include frequencies used for various forms of wireless communication.Blue light has a higher frequency than X-rays: This statement is false. Blue light has a higher frequency than red light, but X-rays have much higher frequencies than both visible light and blue light. Ultraviolet radiation causes common sunburn: This statement is true. Ultraviolet (UV) radiation, specifically UV-B and UV-C rays, can cause sunburn on exposed skin. Overexposure to UV radiation can damage skin cells. A vertical automobile antenna is sensitive to electric fields polarized horizontally: This statement is true. A vertical automobile antenna is designed to receive radio waves that have their electric fields polarized vertically. Gamma rays travel in a vacuum at the same speed as visible light: This statement is false. Gamma rays, which are high-energy electromagnetic radiation, do not travel at the same speed as visible light in a vacuum. X-rays can be produced in transitions involving inner electrons in an atom: This statement is true. X-rays can be generated when electrons undergo transitions from higher energy levels to lower energy levels within an atom. The sun's radiation is most intense in the visible region: This statement is true. The sun emits radiation across a broad spectrum, including radio waves, infrared, visible light, ultraviolet, X-rays, and gamma rays.Learn more about radiations here:
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what force does the water exert (in addition to that due to the atmospheric pressure) on a submarine
Answers
The water that is displaced produces buoyancy, an upward force. As opposed to gravity, which would pull the ship downward, buoyancy acts in the opposite manner.
When the mass of water that the submarine displaces (pushes out of the way) equals the mass of the boat, the submarine (or any boat) can float. The deeper surface is under more pressure. The important thing to keep in mind is that regardless of the depth of the submerged submarine is at, the pressure differential between higher and lower surfaces is always the same since water is incompressible (for all practical purposes here). You mentioned a push, and that contributes to the net buoyancy. To overcome its buoyancy, the submarine must allow water into its ballast chambers. The submarine will need to make modifications while below, adding or "blowing" ballast water to make up for changes in the water's density brought on by temperature or salinity changes, in order to maintain constant depth.
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what fraction of an original element will remain after six half-lives?
Answers
Answer:
1/64
Explanation:
After 1 half-life, 1/2 of the element will remain. Therefore, after 6 half lives, (1/2)^6 of the element will remain. (1/2)^6 = 1^6 / 2^6 = 1/64. So, 1/64 of the element will remain after 6 half-lives.