:) Coal is extracted from a mine at a rate that is proportional to the amount of co that remains in the mine. Show that if R is the amount that remains after t yea R, is the initial amount of coal in the mine and k is a constant then R = R,e -kt Given that after 10 years, 80% of the initial amount of coal remains, find the value of the constant k. (i) (ii) find the time taken for 50% of the initial amount of coal to remain.

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A:

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Q: A large tank is filled to capacity with 2000 L of pure water. Brine containing 0.25 kg salt per…

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A: Sol

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A: GIVEN dpdt =k1200-P P0 =300 CONCEPT SOLVE THE DIFFERENTIAL EQUATION BY VARIABLE SEPARABLE AND BY…

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Q: (c) Coal is extracted from a mine at a rate that is proportional to the amount of coal that remains…

A:

Q: A car traveling with velocity 24 m/s begins to slow down at time t = 0 s with a constant…

A: Part (a) - At t =0,Initial velocity = u = 24 m/sAt time 't',Final velocity = v(t) Acceleration = a…

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A: We have the following differential rate equation :  dc/dt = -kc  ,  k=0.175 And we have to apply…

Q: A tank contains 50 kg of salt and 2000 L of water. A solution of a concentration 0.0125 kg of salt…

A: "Since you have posted a question with multiple subparts, we will solve the first three subparts for…

Q: Suppose you have just poured a cup of freshly brewed coffee with temperature 95°C in a room where…

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Q: A room with a volume of 5000 ft3 is full of air which contains 2% (CO). Pure air containing no (CO)…

A: Given that: Volume = 5000 ft3 Let x(t) denotes the amount of CO in the room at time t. The room…

Q: d) A population of size x is decreasing according to the law dx -x dt 250 where t denotes the time…

A: The solution is given as

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A:

Q: A. What is the limiting value of the temperature of the coffee? lim T(t) 25 t 00 B. What is the…

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Q: 1. Imagine you have two chemicals floating around in a chamber that react in equal quantities to…

A: Given the time integral function is: Tq=∫0qka-xb-xdx For q=a2, evaluate the integral. Let,…

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Q: 6. An object is dropped from the top of a building 200 m high. The object encounters air resistance…

A: Solution :-

Q: b) A population of size x is decreasing according to the law dx -x dt 250 where t denotes the time…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

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Q: An antibiotic is administered intravenouslyinto the bloodstream at a constant rate r. As the drug…

A: Given differential equation model for antibiotic in the bloodstream is  dydt=r-ky, k>0  where k…

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A: Given, dhdt=-kh at t=0 win h=16 m and  t=30 win h=4m

Q: 37. Boyle's Law states that when a sample of gas is compressed at a constant temperature, the…

A: Differentiate  wrt t

Q: contains 60 gallons of pure water. A mixture consisting of 1 pound of salt per gallon is flowing…

A: Given that a tank contains 60 gallons of pure water. A mixture consisting of 1 pound of salt per…

Q: 4) A culture initially has Po number of bacteria. At t = 1 the number of bacteria is measured to be…

A: Given that initially, number of bacteria present in the culture is P0 At t=1, number of…

Q: The population of a town grows at a rate proportional to the population present at time t. The…

A: The solution of above question is written below.

Q: 3 The rate of cooling of a body can be expressed as dT = -k(T – T,), dt Where T is temperature of…

A: It is given that dTdt = -kT-Ta and we know k = 0.154 and Ta = 24 degrees Celsius.

Q: 4. A tank is initially filled with 300 gallon of salt solution containing 1 kg of salt per gallon. A…

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Q: New-cars have a distinctive smell which is caused by the out-gassing of chemicals from the carpet,…

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Q: Imagine you have two chemicals floating around in a chamber that react in equal quantities to make a…

A: From the given function, T(q)=∫0qka-xb-xdx PART A As q→a2 therefore, t=∫0a2k…

Q: Over a period of time, a hot object cools to the temperature of the surrounding air. This is…

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Q: contains a solution of 10 pounds of salt in 60 gallons of water. Water with 1/2 pound of salt per…

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Q: I If f"(x)+ f'(x) + f²(x) = x² be the differential equation of a curve and let P be the point of…

A: Introduction :

Q: A population, P(t), of wild animals is growing proportionally to a function of the initial size, and…

A: What steps to find the size of the population after 5 years when size of population after 1 year is…

Q: Over a period of time, a hot object cools to the temperature of the surrounding air. This is…

A: Answer:   The answer is as shown in below pic :

Q: When a vertical beam of light passes through a transparent medium, the rate at which its intensity I…

A:

Q: At 7 pm on a Wednesday evening. Mr Huxley's water tank was full. The capacity of the tank was 3 000…

A: We will find out the required values.

Q: 3. Boule's Law states that when a sample of gas is compresses at a constant temperature, the…

A:

Q: 4. A certain volume of a ideal gas satisfies PV = kT , where T is the absolute temperature, P is the…

A: Take the derivative with respect to time t and use the concept of derivative of the form  f = uv

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Q: 1. A population of insects at a rate of 200 +10t +0.25t² insects per day (t in days). Find the…

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Q: 5. For high-speed motion through the air-such as a skydiver falling before the parachute is…

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Q: In an experiment, a certain type of bacteria was being added to a culture at the rate of e^(0.03t +…

A:

Q: At time t = 0, a tank contains 20 oz of salt dissolved in 100 gallons of water. Then brine…

A: Given that:                        At t=0, Salt present = 20 oz in 100 gallons of water.Brine…

Q: A disease is spreading through the country. Let x be the number of people infected. Let the constant…

A: Given that, dxdtαxS-x So, the given equation becomes, dxdt=kxS-x For some constant K.…

Q: 1. For a substance D, the time rate of conversion is proportional to the square root of the amount x…

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Q: The air in a room with volume 180 m3 contains 0.1% carbon dioxide initially. Fresher air with only…

A: Let % of CO2 at time t be ptAmount of CO2 in room=volume of room×pRate of change of…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.