According to Newton's law of cooling, thetemperature of a body changes at a rate proportionalto the difference between the temperature of thebody and the temperature of the surroundingmedium. Thus, if Tm is the temperature of themedium and T = T(t) is the temperature of thebody at time t, thenT'= -k(T - Tm)where k is a positive constant.In this exercise, you will verify thatT = Tm+ (To-Tm)e -kt is a solution to thedifferential equation. Here To denotes the initialtemperature T(0) of the body.(a) First substitute the above expression for T intothe left side of the differential equation. In otherwords, compute T'.Use an underscore "_" to write the subscripts on Toand ImT'=(b) Next substitute the above expression for T intothe right side of the differential equation. In otherwords, compute -k(T – Tm).--k(T - Tm)(c) Are your answers in parts (a) and (b) equal?NoYes

Answered: Image /qna-images/answer/fb58ea57-21e0-4a24-b78c-6ab02d8f184e.jpg

Answered: According to Newton's law of cooling,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A tank is filled with 200 l of salt solution containing Qo kg of salt at time t = 0. Another salt solution with 1/4 kg of salt per liter flows into the tank at a rate of 6 lt / min. By mixing, a homogeneous water-salt mixture is obtained continuously in the tank and the mixture leaves the tank at the same speed. Find the expression Q (t) which gives the amount of salt present in the tank at time t.
Newton's law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) dT 1 and the temperature of the body. That is, = K[M(t) - T(t)], where K is a constant. Let K = 0.04 (min) and the temperature of the medium be constant, dt M(t) = 292 kelvins. If the body is initially at 360 kelvins, use Euler's method with h = 0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes.
According to Newton's Law of Cooling, the temperature T of an (as a function of time t) is given by T(t) = T, + D,e, where k is a positive constant, and D, is the difference between T(0) (the temperature at time t = 0) and T, (the temperature of the surroundings). If a bowl of soup with temperature 210 degrees Fahrenheit is set down in a room with temperature 65 degrees, after how many minutes will the soup's temperature reach 100 degrees? = 0.05] 10. [for this soup, assume k
Because of heavy rainfall, the water level in a reservoir is rising at a rate of g(t) feet per hour, where t = 0 corresponds to noon. What does g(t) dt represent?
Let y(t) be the height (in meters) after time t (in seconds) of a rover that is landing on some other planet with the help of a parachute. Taking into account gravity and air resistance, physicists suggest that the air resistance is proportional to velocity. Further data suggests that the fall can be modeled by y"=-4-3y'. If the parachute is released at a height of 600m with an initial velocity of 0m/s, what is its height after t seconds? What is its terminal velocity? y(t) = The terminal velocity is m/s. m/s
Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). We let (t) denote time, T(t) be the temperature of the object at time (t), (T0 = T(0)) (the object’s temperature at (t = 0)), and Tomg be the constant ambient temperature. Set u(t) = T(t) - Tomg and show from Newton’s law of cooling that (u) satisfies the initial value problem (*) u’(t) = ku(t), u(0) = T0 - Tomg for a constant (k). Solve the initial value problem (*), i.e., find u(t)) (expressed in terms of (k), T0, and Tomg . Find T(t).
Let y(t) be the height (in meters) after time t (in seconds) of a rover that is landing on some other planet with the help of a parachute. Taking into account gravity and air resistance, physicists suggest that the air resistance is proportional to velocity. Further data suggests that the fall can be modeled by y"=-5-3y'. If the parachute is released at a height of 600m with an initial velocity of 0m/s, what is its height after t seconds? What is its terminal velocity? y(t) = The terminal velocity is Submit m/s. m/s
A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a(t) = 90t, at which time the fuel is exhausted and it becomes a freely "falling" body. Eighteen seconds later, the rocket's parachute opens, and the (downward) velocity slows linearly to -11 ft/s in 5 seconds. The rocket then "floats" to the ground at that rate. (a) Determine the position function s and the velocity function v (for all times t). 4512 if 0 sts 3 405 if 3 26 15r3 if 0 sts 3 405 if 3 26
Let y(t) be the height (in meters) after time t (in seconds) of a rover that is landing on some other planet with the help of a parachute. Taking into account gravity and air resistance, physicists suggest that the air resistance is proportional to velocity. Further data suggests that the fall can be modeled by y"=-4-3y'. If the parachute is released at a height of 600m with an initial velocity of 0m/s, what is its height after t seconds? What is its terminal velocity? y(t) The terminal velocity is m/s. m/s
Let y(t) be the height (in meters) after time t (in seconds) of a rover that is landing on some other planet with the help of a parachute. Taking into account gravity and air resistance, physicists suggest that the air resistance is proportional to velocity. Further data suggests that the fall can be modeled by y=-4-3y. If the parachute is released at a height of 600m with an initial velocity of Om/s, what is its height after t seconds? What is its terminal velocity? The terminal velocity is m/s. m/s
Calculate G(16), where dG/dt =t-1/2 and G(9) = -5.

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