Linear DDEs: Problem 1 Previous Problem List Next Results for this submission En
ID: 2885224 • Letter: L
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Linear DDEs: Problem 1 Previous Problem List Next Results for this submission Entered Answer Preview Result incorrect The answer above is NOT correct (1 point) 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 210 degrees Fahrenheit when freshly poured, and 1.5 minutes later has cooled to 194 degrees in a room at 72 degrees, determine when the coffee reaches a temperature of 154 degrees. The coffee will reach a temperature of 154 degrees in minutes.Explanation / Answer
Newton's law of cooling: T(t)= Ta+(T0-Ta)e-kt
Ta=72 ,T0=210 ,T(1.5) =194
T(t)= 72+(210-72)e-kt
=>T(t)= 72+ 138e-kt
=>T(1.5)= 72+(210-72)e-k*1.5
=>194= 72+(210-72)e-1.5k
=>194= 72+ 138e-1.5k
=>138e-1.5k=122
=>e1.5k=(138/122)
=>1.5k=ln(69/61)
=>k=(2/3)ln(69/61)
=>T(t)= 72+ 138e-((2/3)ln(69/61)) t
when temperature reaches 154 degrees
T(t)=154
=>72+ 138e-((2/3)ln(69/61)) t=154
=>138e-((2/3)ln(69/61)) t=154-72
=> 138e-((2/3)ln(69/61)) t=82
=> e((2/3)ln(69/61)) t=138/82
=>((2/3)ln(69/61)) t=ln(69/41)
=>t=(3/2)(ln(69/41)/ln(69/61))
=>t=6.3359971364185106151245404447221
The coffee will reach a temperature of 154 degrees in 6.3 minutes
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