Exponential and Logarithmic Models

Question

A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit. Give answers accurate to at least 2 decimal places.

(a) If the temperature of the turkey is 141 Fahrenheit after half an hour, what is its temperature after 45 minutes?

   Fahrenheit

(b) When will the turkey cool to 100 Fahrenheit?

   hours.

Answer 1

Let T be the temperature of the turkey at any time, and T_m be the constant ambient temperature in our case

T_m = 75.

Then according to the empirical Newton's Law of Cooling the rate at which the turkey's temperature changes is proportional to the difference between the turkey's temperature and the temperature of the surrounding medium

dT / dt = k [ T - T_m ]

dT / [ T - T_m ] = k dt

∫ dT / [ T - T_m ] = ∫ k dt

ln | T - T_m | = k t + c

|T - T_m| = e( k t + c)

T - T_{m =} λ e^kt

T = T_m + λ e^kt

T = 75 + λ e^kt But we are given that at t = 0 T= 185

185 = 75 + λe⁰ That is λ = 110.

T = 75 + 110 e^kt . But there is one more constant to be determined and that is the value of the constant k.

Also we are given that at t = .5 the temperature is T= 141. Thats translates to the equation

141 = 75 + 110 e^k/2 66/ 110 = e^k/2 k = -1.0216. Now the equation of the law is completely defined as

T = 75 + 110 e^-1.0216t

Then the temperature 45 minutes later is

T = 75 + 110 e^-1.0216(.75) = 126.12

The temperature will reach 100 when

100 = 75 + 110 e^-1.0216t

110 e^-1.0216t = 25

e^-1.0216t = 5 / 21

t = 1.40 hours

That is one hour and 40% of an hour later .

In other words 1 hour and 24 minutes later