Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to thetemperature difference between the object and its surroundings. This can be modeled by the differentialequation =k(TA), where T is the temperature of the object after t units of time have passed, A isdTdtthe ambient temperature of the object's surroundings, and k is a constant of proportionality.Suppose that a cup of coffee begins at 179 degrees and, after sitting in room temperature of 62 degrees for14 minutes, the coffee reaches 170 degrees. How long will it take before the coffee reaches 160 degrees?Include at least 2 decimal places in your answer.minutes

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Answered: Newton's Law of Cooling tells us that…

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 piece of chicken is boiled until its internal temperature reaches 165 °F. The chicken is then transferred to a refrigerator held at 49 °F to cool down. The internal temperature of the chicken at time t is given by T(t), and the temperature obeys Newton's law dT = -k(T - Tm) where Tm is the temperature of the refrigerator and k > 0 is a constant. After 5 minutes in the refrigerator, the internal temperature of the chicken is 155 °F. How long will it take for the temperature of the chicken to reach 70 °F?
For a rate r (in decimal form) that is compounded n times per year, the effective rate reff (in decimal form) is reff=(1+r/n)n-1 For a rate r (in decimal form) that is compounded continuously, the effective rate reff (in decimal form) is reff=er-1 Use the above equations to complete the table showing the effective rates for nominal rates of r=4%, r=7%, r=9 1/2% (Round your answers to five decimal places.) r = 4% Number ofcompoundingsper year 4 12 365 Continuous Effective yield r = 7% Number ofcompoundingsper year 4 12 365 Continuous Effective yield r = 9 1/2% Number ofcompoundingsper year 4 12 365 Continuous Effective yield
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
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).
At 7 pm on a Wednesday evening. Mr Huxley's water tank was full. The capacity of the tank was 3 000 litres. Unfortunately, the tap on the tank was leaking in such a way that the change in volume at any time (r) hours was proportional to the volume (V) of the tank. This means that AP =-kV . dt Show that V= Ve is a solution of this equation. Given that the volume of the tank after 3 hours is 1900 litres, show that k = 0.1523 correct to 4 decimal places. (iii) By the time Mr Huxley discovered that the tank was leaking, there were only 250 litres of water remaining. At what time and on which day did Mr Huxley discover the leak (Answer correct to the nearest minute)?
Near the surface of the earth, a ball (with p = 0.1) is released from rest and its flight through the air offers resistance that is proportional to its velocity. How long will it take the ball to reach one-half of its terminal velocity? How far will it travel during this time?
According to Newton's law of cooling, the temperature of a body changes at a rate proportional to the difference between the temperature of the body and the temperature of the surrounding medium. Thus, if Tm is the temperature of the medium and T = T(t) is the temperature of the body at time t, then T'= -k(T - Tm) where k is a positive constant. In this exercise, you will verify that T = Tm+ (To-Tm)e -kt is a solution to the differential equation. Here To denotes the initial temperature T(0) of the body. (a) First substitute the above expression for T into the left side of the differential equation. In other words, compute T'. Use an underscore "_" to write the subscripts on To and Im T'= (b) Next substitute the above expression for T into the right side of the differential equation. In other words, compute -k(T – Tm). - -k(T - Tm) (c) Are your answers in parts (a) and (b) equal? No Yes

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