ccording to Newton\'s Law of Cooling, if a body with temperature Upper T 1T1 is
ID: 3112664 • Letter: C
Question
ccording to Newton's Law of Cooling, if a body with temperature Upper T 1T1 is placed in surroundings with temperature Upper T 0T0, different from that of Upper T 1T1, the body will either cool or warm to temperature T(t) after t minutes, where T(t)equals=Upper T 0T0plus+(Upper T 1T1minusUpper T 0T0)e Superscript negative ktekt. A cup of coffee with temperature 160degrees°F is placed in a freezer with temperature 0degrees°F. After 15 minutes, the temperature of the coffee is 47degrees°F. Use Newton's Law of Cooling to find the coffee's temperature after 20 minutes.
Explanation / Answer
According to Newton's law of cooling ,
T(t) = T0 + ( T1 - T0 ) e-rt
Where , T(t) = temp of the body after t minutes
T0 = temp of the surrounding
T1 = initial temp of the body
r = constant and t= time in minutes
Now, putting the values in the equation we get ,
After t =15 mins :
470F = 00F + ( 1600F - 00F ) e-15r
=> e-15r = 47/160
taking log both sides we get , -15r (ln e) = ln (47/160) => r = - { ln (47/160)/ ln e } / 15 = 0.082 ....... (a)
After t = 20 mins :
T(20) = 00F + ( 1600F - 00F ) e-20r
=> T(20) = 160 e-20r
Putting value of r from eq (a) , we get :
=> T(20) = 160 e- 20(0.082) = 160 * 0.195 = 31.25
Hence , the coffee's temperature after 20 minutes will be 31.25 0F .
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