Newton\'s Lawof Cooling states that the rate of change of thetemperature u of an
ID: 2938309 • Letter: N
Question
Newton's Lawof Cooling states that the rate of change of thetemperature u of an object is
proportionalto the difference between uand the (constant)temperature Tof thesurrounding
medium.
a) Give a mathematicalformulation of this statement in the form of a differentialequation.
b) A thermometer istaken from a room where the temperature is 72oFto the outside where the
temperature is32oF. After3 minutes, the thermometer reads 65oF. Whatwill the themometer
read after it has been outside for5 minutes?
Explanation / Answer
Newton's Lawof Cooling states that the rate of change of thetemperature u [DU/DS]of an object is
proportionalto the difference between uand the (constant)temperature T of the surrounding
medium.[U-T].... S ISTIME
a) Give a mathematicalformulation of this statement in the form of a differentialequation.
DU/DS=K*[U-T].................................1
WHERE K IS THEPROPORTIONALITY CONSTANT
b) A thermometer istaken from a room where the temperature is 72oFto the outside where the
temperature is32oF. After3 minutes, the thermometer reads 65oF. Whatwill the themometer
read after it has been outside for5 minutes?
FROM EQN.1
DU/[U-T]=KDS
INTEGRATING
LN[U-T]=KS
U-T=C*EKS..............................2
WHERE C IS A CONSTANT OFINTEGRATION
WE HAVE ....T=32 F.......AT STARTWHEN S=0 TIME,WE HAVE U=72 F
PUTTING IN EQN.2 WEGET
72-32=C=40
HENCE EQN.2 BECOMES
U-32=40EKS..........................3
U-32=40E-0.0641S..........................4
AT S= 5 , WE GET
U-32=40E-0.0641*5=29.03
U=61.03 OR SAY 61 DEG.F.