As part of his summer job at a resturant, Jim learned to cook up a big...

  

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As part of his summer job at a resturant, Jim learned to cook up a big pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100 degrees C, and the fridge was not powerful enough to accomodate a big pot of soup if it was any warmer than 20 degrees C). Jim discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5 degrees C) and stirring occasionally, he could bring the temperature of the soup to 60 degrees C in ten minutes. How long before closing time should the soup be ready so that Jim could put it in the fridge and leave on time? Let x+ denote the temperature of the soup at time t. The initial tempearture of the soup is x0 = 100 degrees C. The ambient temperature, i.e., the temperature of the sink of cold water is 5 deggrees C. According to Newton's law of cooling, the evolution of Xt is governed by the following differential equation: dx dt = -k (x-5), X0=100. where k > 0 is a constant that characterizes the cooling process. This differential equation asserts that the rate of change of the temperature of the soup is negatively proportional to the difference in its temperature and the temperature of the ambient environment. That is, the greater the difference between the temperature of the soup and the sink of cold water, the faster the soup cools. (a) Solve this differential equation. (b) We are told that X10 = 60 degrees C. Use this information to compute the parameter k. (c) Find the time t at which xt = 20 degrees C. As part of his summer job at a resturant, Jim learned to cook up a big pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100 degrees C, and the fridge was not powerful enough to accomodate a big pot of soup if it was any warmer than 20 degrees C). Jim discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5 degrees C) and stirring occasionally, he could bring the temperature of the soup to 60 degrees C in ten minutes. How long before closing time should the soup be ready so that Jim could put it in the fridge and leave on time? Let x+ denote the temperature of the soup at time t. The initial tempearture of the soup is x0 = 100 degrees C. The ambient temperature, i.e., the temperature of the sink of cold water is 5 deggrees C. According to Newton's law of cooling, the evolution of Xt is governed by the following differential equation: dx dt = -k (x-5), X0=100. where k > 0 is a constant that characterizes the cooling process. This differential equation asserts that the rate of change of the temperature of the soup is negatively proportional to the difference in its temperature and the temperature of the ambient environment. That is, the greater the difference between the temperature of the soup and the sink of cold water, the faster the soup cools. (a) Solve this differential equation. (b) We are told that X10 = 60 degrees C. Use this information to compute the parameter k. (c) Find the time t at which xt = 20 degrees C.