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Solving for p we get t e po po t e po po t p 3 4 1 3

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Solving for P we get t e Po Po t e Po Po t P 3 ) 4 ( ) 1 ( 3 ) 4 ( ) 1 ( 4 ) ( - - - - - - - - = 4.13 Exercises Newton’s Law of Cooling/Warming 1. A thermometer reading 100 0 F is placed in a pan of oil maintained at 10 0 F. What is the temperature of the thermometer when t=20 sec, if its temperature is 60 0 F when t = 8 sec? 2. A thermometer is removed from a room where the air temperature is 60 0 F and is taken outside, where the temperature is 10 0 F. After 1 minute the thermometer reads 50 0 F. What is the reading of the 111
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thermometer at t=2 minutes? How long will it take for the thermometer to reach 20 0 F . 3. Water is heated to a boiling point temperature 120 0 C. It is then removed from the burner and kept in a room of 30 0 C temperature. Assuming that there is no change in the temperature of the room and the temperature of the hot water is 110 o C after 3 minutes. (a) Find the temperature of water after 6 minutes (b) Find the duration in which water will cool down to the room temperature? Population Growth and decay 4. A culture initially has P o number of bacteria. At t=1 hour, the number of bacteria is measured to be 2 3 P 0 . If the rate of growth is proportional to the number of bacteria P(t) present at time t, determine the time necessary for the number of bacteria to triple. 5. Solve the logistic differential equation: 0 N ) 0 ( N , 0 t , N ) k N 1 ( o r dt dN = - = 6. Insects in a tank increase at a rate proportional to the number present. If the number increases from 50,000 to 100,000 in one hour, how many insects are present at the end of two hours. 7. It was estimated that the earth’s human population in 1961 was 3,060,000,000. Assuming the population increases at the rate of 2 112
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percent, find the earth’s population in 1996 using model of population growth (1.8). Check this number with the actual population of the earth available from authentic sources. 113
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8. A breeder reactor converts relatively stable uranium 238 into the isotope plutonium 239. After 30 years it is determined that 0.022% of the initial amount N 0 of plutonium has disintegrated. Find the half-life of this isotope if the rate of disintegration in proportional to the amount remaining. 9. The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life of 4 hours. If 1 gram of lead is present initially, how long will it take for 80% of the lead to decay? 10. Solve the model obtained in Exercise 32 of chapter 1. 11. In the 1950 excavation at Nippur, a city of Babylonia, charcoal from a roof beam gave a count of 4.09 dis/min/g. Living wood gave 6.68 disintegrations. Assuming that this charcoal was formed during the time of Hammurabi’s reign, find an estimate for the likely time of Hammurabi’s succession. Mixture of Two Salt Solutions 12. A tank with a capacity of 600 litres initially contains 200 litres of pure water. A salt solution containing 3 Kg of salt per litre is allowed to run into the tank at a rate of 16 lit/min, and the mixture is then removed at a rate of 12 lit/min. Find the expression for the number of Kilograms of salt in the tank at any time t. 114
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Solving for P we get t e Po Po t e Po Po t P 3 4 1 3 4 1 4...

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