Temperature varies - Calculus

Question

The temperature of a hard boiled egg varies according to Newton's Law of Cooling: dT/dt= -k(T-A), where T is the temperature of the egg, A is the room temperature, and k is a positive constant. If the egg cools from 100°C to 90°C in 1 minute at a room temperature of 70°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes.  

50

76

79

82

Answer 1

The solution to the differential equation is

 

T(t) = A + (T0 - A)*e^(-kt)

 

T0 = initial temperature of body

 

We need a value for k.

 

T goes from 100 deg to 90 deg in 1 minute.

 

A = 70 deg, T0 = 100 deg, T(1 min) = 90 deg

 

90 = 70 + (100 - 70)*e^[-k(1)]

 

k = - ln(20/30) = 0.405465 deg/min

 

T(4 min) = 70 + (100 - 70)*e^[-(0.405465)(4)]

 

T(4 min) = 75.93 degrees C, i.e. about 76 degrees C