Newton's Law of Cooling is based on the principle that the rate of change of temperature...

Newton's Law of Cooling is based on the principle that the rate of change of temperature y'(t) of a body in an environment with ambient temperature A is proportional to the difference b/w the temperature y(t) of the body and the ambient temperature A, so that y'=k(A-y) for some constant k.

(1) Sketch the direction field for the equation y'=K(A-y) for k=1/10 and A=70degrees. Then sketch several solution curves based on starting values of y(0) both greater than and less than A. Discuss the implication of your graph in general, and as it relates to the long-term limiting temperature of a cup of hot coffee initially at 190degreeF and a cup of iced coffee initially at 40degreesF in a room with ambient temperature 70degreesF.

Expert Solution

Colby Messinger answered 2 years ago

Newton's Law of Cooling tells us that the rate of change of the temperature of an...

Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation d T d t = k ( T − A ) , where T is the temperature of the object after t units of time have passed, A is the ambient temperature of the object's surroundings, and k is a constant of proportionality....

Newton's law of cooling states that the temperature of an object changes at a rate proportional...

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 205 degrees Fahrenheit when freshly poured, and 1 minutes later has cooled to 190 degrees in a room at 64 degrees, determine when the coffee reaches a temperature of 150 degrees. The...

2. According to Newton’s Law of Cooling, the rate of change of the temperature of an...

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differential equation 12. Newton's law of cooling states that the temperature of an object changes at...

differential equation 12. Newton's law of cooling states that the temperature of an object changes at a rate pro-portional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 200F when freshly poured, and 1 min later has cooled to 190F in a room at 70F, determine when the coffee reaches a temperature of 150F.

Newton's Law of Cooling states that the rate at which and objects cools is directly proportional...

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10. Newton’s law of cooling states that the temperature of an object changes at a rate...

10. Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between the temperature of the object an the temperature of its surroundings. (a) Write a differential equation for the temperature of the object at any time. b) State the dimensions of the constant of proportionality, and its SI units. C) Plot the direction field for the DE and find the limiting behaviour of solutions. If this limit depends on the...

According to Newton’s law of cooling, the temperature of a body changes at a rate pro-...

According to Newton’s law of cooling, the temperature of a body changes at a rate pro- portional to the difference between its temperature and that of the surrounding medium (the ambient temperature), dT dt = −k(T − Ta) with T(0) = T0 where T is the temperature of the body in oC, t is time in min, k = 0.019 min−1 is the proportion- ality constant, and Ta = 20 oC is the ambient temperature. The analytical solution for this...

Newton's Law of Cooling: Boil water in a cup and cool in a room at 32∘C....

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2. A coroner uses a formula, derived from Newton's Law of Cooling, to calculate the elapsed...

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An ODE for Newton’s law of cooling, ??/?? + ?? = ??(?) relates indoor temperature changes...

An ODE for Newton’s law of cooling, ??/?? + ?? = ??(?) relates indoor temperature changes u(t) to outdoor temperature changes modeled by the function ?(?) = ? + ? cos(??) + ? sin(??). k is a constant that measures how well a building is insulated and t is time in hours. Suppose your air-conditioner is broken. You want to find out how outdoor temperature effects the temperature in your house by doing the following: (a) Find a model for...

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