(a) The wate of change of temperature of the body i directly proportional to the difference between the temperature of the body and the ambient temperature (b ) Mathematically OT o ( T -TO ) At the sign of proportionality can be removed by Introducing a constant x ( Lambda ) 20 24 = -X (T-79 ) a minus ( - ) sign is Introduced on the right side If the equation , because on the left side AT (T-Ta ) (it ) at where , dT - rate of change of temperature 56 body 2 - constant 2 0 7 - temperature of body at any instant To - ambient temperature( c) ( 1) using equation: JI - ALT- To ) de dT -Adt ( T- TA ) at Himat - > T T di - - adt (T - TA ) (n ( T- T.) - In ( T - Te ) - ( ( 7. - 7. ) = _ nt T - Ta 7 . - To T - To = ( T . - To ) C Ta Ta + ( T. - TO) To - Ta (c 16 - z(+) To - Ta (n z () . _ at at too, when body is dillovered can be replaced by to and further time can be bepresented by I .+ 2cplaid by ta + Z (n 2 (+12 - > (ta+ ( )After someone dies their body temperature decreases due to heat-exchange with the environment, even- tually reaching the ambient temperature. Suppose that the body temperature at the time of death is To- (a) Write down a word equation for the heat-transfer processes occurring following the death of an indi- vidual. (b) Convert your word equation into a mathematical model (i) Carefully state any assumptions that you make. (ii) Define all symbols in your model. (c) (i) Find the temperature of the body (T) as a function of time. (ii) Assume that the individual dies at time t = 0. The body is discovered at time t = ta and that time after the body is discovered is represented by r. Show that your solution can be written in the form In Z (t) = -> (ta +T), where you are to identify the variables 2 and A in terms of the variables in in your model. (d) Explain the process by which the time-of-death can be estimated given temperature measurements made after the body is discovered. (e) At time t = ta a body is found. The ambient temperature is 24"C. After discovery the temperature of the body is measured. These measurements are shown in table 1. Given that a typical body temperature for a living human is 37 degrees centigrade and that the body was found at 2pm estimate when the individual died. Time (minutes) body temperature (C) 0 36.0 60 35.8 120 35.5 180 35.3 240 35.1 300 34.9 Table 1: Body temperature of a dead individual as a function of the time after the body was discovered