Newtons law of cooling expresses the relationship between th

Newton\'s law of cooling expresses the relationship between the temperature of a cooling object y and the time t elapsed since cooling began. This relationship is given by = ae ^kt + c, where c is the temperature of the medium surrounding the cooling object, a is the difference between the initial temperature of the object and the surrounding temperature, and k is a constant related to the cooling object. The initial temperature of a liquid is 160 ^degree F. What it is removed from the heat the temperature of the liquid after 15 minutes. Alex likes his coffee at a temperature of 135 ^degree. If he pours a cup of 170 ^degree F in a 75 ^degree F room and waits 5 minutes before drinking, will his coffee be too or too cold? Explain. For Alex\'s cup, k = 0.34.

Solution

y = ae^(-kt) + c where is the temperature , a is the difference between the intial temp. of the object and surrounding temp, and k is a constant

a) a = 160 -76 = 84F k =0.23 t = 15min., c= 76 degF Find y

y = 84e^(-0.23*15) +76 = 78.67 degF

b) y = 135 ; a= 170 -72 = 98 degF; k = 0.34 ; c= 72 degF , Find t

y = ae^(-kt) + c

135 = 98e^(-0.34*t) + 72

0.6428 = e^(-0.34*t)

Taking natural log on both sided:

ln(0.6428) = -0.34t

t = 1.3 minutes