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# Newtons law of cooling

Sir Isaac Newton investigated the cooling process of water temperature and showed that the temperature change is exponential in time and can be predicted by
Tdiff=T0e-kt
Where Tdiffis the difference in temperature between the water and the air in the room,
T0 is the initial difference in the temperatures (Tt=0 – T room)
So if T is temperature of he water any time, the equation can be re-written as
T-T room=(Tt=0-Troom)e-kt

Where k is constant of proportionality and t is time.

Use the equation to solve this problem:

A hard boiled egg at 98°C is put in a sink of 18°C water to cool. After 5 minutes, the eggs temperature is found to be at 38°C. Assuming that the water has not warmed up, how much longer will it take the egg to reach 20°C?

### 2 Answers by Expert Tutors

Glenn M. | Math and Science TutorMath and Science Tutor
4.9 4.9 (401 lesson ratings) (401)
1
use the information at 5 minutes to calculate the value for k (constant).

Then use the value for k to determine the time it will take to reach the final temp (20 C)

38=18+(98-18)e-k(5)
20=80e-k(5)
ln(20/80)=-k(5)
k=.2773

20=18+(98-18)e-.2773(t)
2=80e-.2773(t)
ln(2/80)=-.2773(t)
t=13.303

Hence 13.3-5=8.3

So it will take an additional 8.3 minutes for the egg to reach 20C
Philip P. | Effective and Affordable Math TutorEffective and Affordable Math Tutor
5.0 5.0 (428 lesson ratings) (428)
0
(T-Twater) = (T0-Twater)e-kt

The first step is to use the data about how much the egg has cooled after 5 minutes to find the value of k, which is unknown.

(38-18) = (98-18)e-5k
20 = 80e-5k
20/80 = 1/4 = e-5k
ln(1/4) = -5k
ln(1/4)/(-5) = k
0.277 ≅ k

Now that we have the value of k, we can find how much longer it will take the egg to cool to 20oC.  In this case, the egg has already cooled to 38oC, so the starting or initial temp is 38oC, not 98oC.

(20-18) = (38-18)e-(0.277)t
2 = 20e-(0.277)t
2/20 = 1/10 = e-(0.277)t
ln(1/10) = -(0.277)t
ln(1/10)/(-0.277) = t
8.3 minutes ≅ t