The following exercises deal with Newton\'s Law of Cooling, which can he express

Question

The following exercises deal with Newton's Law of Cooling, which can he expressed using the formula t = k ln (T_0 - C/T - C) Here t is the time it will take for an object to cool to a temperature T. T_0 is the initial temperature, C is the temperature of the surrounding air, and k is a constant. Coffee Temperature Find the time required for a hot cup of coffee with an initial temperature of 190 degree F to cool to a temperature of 80 degree F if the temperature of the surrounding air is 70 degree F and k = 20 (for time measured in minutes). Ice Tea Temperature A freshly brewed pitcher of tea, currently at a temperature of 175 degree F, is placed in a refrigerator that has a constant temperature of 40 degree F. Find the time required for the tea to reach a temperature of 45 degree F given that k = 50 (for time measured in minutes) Body Temperature Inspector Magill is called to the scene of a murder. The victim is lying in a room that has a constant temperature of 75 degree F. The victim's body temperature is currently 81.2 degree F, down from the normal body temperature of 98.6 degree F. After considering the height and weight of the victim, Magill arrives at a value of k = 10 (for time measured in hours). How long has the victim been dead? Body Temperature Revisited After further consideration (see Exercise 75), Inspector Magill decides that he is not sufficiently confident in his k value to fix the time of death To determine k, he notes that the victims body temperature at 3:00 PM. Wednesday is 81.2 degree F and that by 4:00 am Thursday it has dropped to 77.0 degree F. Assuming that the body

Explanation / Answer

t = kln(To - C)/(T - C)

73) To = 190 F ; C = 70 ; k = 20 ; T = 80 F

t = 20ln(190 - 70)/(80 -70)

= 20ln(120/10)

= 49.69 minute

75) C = 75 F ; To = 98.6 F ; T = 81.2 F ; k =10

ln = 10ln(98.6 -75)/(81.2 - 75)

= 10ln(23.6/6.2)

= 13.36 hours

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