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6、hematics A detective discovers a murder victim in a hotel room at 9 00am one morning The temperature of the body is 80 0 F One hour later at 10 00am the body has cooled to 75 0 F The room is kept at a constant temperature of 70 0 F Assume that the victim had a normal temperature of 98 6 F at the tim
7、e of death We ll use diff erential equations to fi nd the time the murder took place Let u t be the temperature of the body after t hours By Newton s Law of Cooling we have the diff erential equation du dt k u 70 where k is a constant to be determined We ll solve the diff erential equation and get a
8、 formula for u t 1 Multiply both sides by dt to get a diff erential form of the equation Now do some easy algebra to get the variable u on the same side as the du Leave the k where it is 2Integrate both sides of the equation Integrate the right side with respect to t and the left with respect to u Y
9、ou can combine the integration constants into one C on the right side 3Solve for u as a function of t Your function will involve the constants k and C 4Take t 0 when the body was found at 9 00am Plug in t 0 and u 80 0 F and solve for C It s easier to solve for A eCand use this in your formula 5Plug
10、in t 1 and u 75 0 F and solve for k This ll take some log tricks 6Set u 98 6 F and solve for t At what time did the murder take place Spread of a Rumor The Xylocom Company has 1000 employees On Monday a rumor began to spread among them that the CEO had suddenly moved to Brazil It is reasonable to as
11、sume that the rate of the spread of the rumor is proportional to the number of possible encounters between employees who have heard the rumor and those who have not Let y y t be number of employees who have heard the rumor after t days 1Explain why the number of possible meetings between employees w
12、ho have heard the rumor and those who have not equals y 1000 y 2 Write a diff erential equation that describes this model of the spread of a rumor Remember that is proportional to means is some constant k times 3 Proceed as in the cooling body problem to solve the diff erential equation for y t You
13、will need to use the method of partial fractions Your answer should involve two constants the proportionality constant k and a constant C from integrating As in part 2 of the previous problem you can combine the integration constants into one C on the right side 4At the very beginning 50 people had
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