A metal bar at a temperature of 100 degreess is placed in aroom at a constant te

Question

A metal bar at a temperature of 100 degreess is placed in aroom at a constant temperature of 0 degrees F. If after 20 minutesthe temperature of the bar is 50 degrees F, find anexpression for the temperature of the bar at any time. If possible use differential equ. to solve A metal bar at a temperature of 100 degreess is placed in aroom at a constant temperature of 0 degrees F. If after 20 minutesthe temperature of the bar is 50 degrees F, find anexpression for the temperature of the bar at any time. If possible use differential equ. to solve If possible use differential equ. to solve

Explanation / Answer

Newton's law of cooling states that the rate of change in thetemperature of an object is proportional to the difference betweenthe object's temperature and the temperature of its surroundings.Let T[t] denote the temperature at time t. Then the differentialequation we must solve is:    T ' [t] = -k(T[t] - 0), where k is a yet tobe determined constant This differential equation is easily solved by separation ofvariables with the initial condition that T[0] = 100 toobtain: T[t] = 100e-kt To find k plug in t = 20 and set this expression equal to 50degrees. Do the algebra and k = log[2]/20. Plug this into thesolution and simplify to the obtaint the complete solutionof: T[t] = 25*22-t/20 To find k plug in t = 20 and set this expression equal to 50degrees. Do the algebra and k = log[2]/20. Plug this into thesolution and simplify to the obtaint the complete solutionof: T[t] = 25*22-t/20

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