Question

You happen to find yourself employed as an expert mathematical consultant for a new Australian TV show called "Numer4ls". The producer wants an episode in which the lead character uses his mathematical skills to solve a murder mystery by accurately determining the time of death of the victim, which in this case will be made more complicated by a varying ambient temperature, and will therefore involve a lot of maths written in liquid chalk on glass panels, and some computer simulations You will use Newton's law of cooling, which states that the rate of heat loss of an object is proportional to the difference between the ambient temperature and the object's temperature (under the assumptions of steady convection, a uniform heat distribution, and the object not affecting the ambient temperature). Therefore t T (t) k (A (t)-T (t)) where T(t) is the object's temperature, A(t) is the varying ambient temperature, and k is a constant of proportionality (degrees/hour/degree), which we take to be k 1/4 This particular episode involves the deceased being found outside on a cold and windless winter's afternoon, having been murdered during the middle of the night (at time t0).Assuming that the ambient temperature follows a sinusoidal pattern, ranging from-8 deg C overnight to +8 deg C during the day, you determine that where t is measured in hours and its coefficient 1/4 is the frequency of the fluctuation, giving a period of approximately 25 hours. (A coefficient of z/12 would give a more realistic period of 24 hours, but we use this approximation to make the numbers easier for you.) The script writer is a suitably macabre looking chap who wants you to tell him what temperature the deceased would be found at after 15 hours, so he can incorporate this into the story Since Newton's law of cooling (1) is linear and first order, it could be solved using the Integrating Factor method. However, since it is also a constant coefficient ODE, it can be solved using the method for constant coefficient ODEs, which is what you must do here. (a) First solve the homogeneous equation and enter your solution below. Note: You must use a capital "B" as your arbitrary constant. Tht) (b) Next use the method of undetermined coefficients to find a particular solution to (1). 自励 Tp(t) = | (c) using the fact that (healthy) living humans maintain a constant body temperature of 37 deg C (approximately), determine the value of your arbitrary constant B (d) Hence determine the temperature of the body 15 hours after death. Note 1: You must round your answer to 2 decimal places, because any more would be too long for TV air time. Note 2: Do not enter the units, just the number. T= |数字

Answer 1

Given To31 (b-TP(t)*T(t) have : we ve Now aftex has, TS89