Newton\'s Law of Cooling states that the rate of cooling of an object is proport
ID: 2882494 • Letter: N
Question
Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose t is time, T is the temperature of the object, and Ts is the surrounding temperature. The following differential equation describes Newton's Law
dT/dt=k(TTs)
where k is a constant.
Suppose that we consider a 90Ccup of coffee in a 24C room. Suppose it is known that the coffee cools at a rate of 2C/min. when it is 70C. Answer the following questions.
1. Find the constant k in the differential equation.
Answer (in per minute): k=-2/46 (correct)
2. What is the limiting value of the temperature?
Answer (in Celsius): T=24 (correct)
I need help with 3.
3. Use Euler's method with step size h=2 minutes to estimate the temperature of the coffee after 10 minutes.
Answer (in Celsius): T(10)
Explanation / Answer
Solution for 3rd part:
dT/dt = k (t - 24)
dT/dt = (-1/23)(t - 24) = -0.04374*(t-24)
given that : step size h=2 minutes ,T = 90
T = 90, let's write T' for dT/dt
T' = -0.04347*(90-24) = -2.8695
With h = 2, deltaT = -2.8695*2 = -5.7391
Therefore T = 84.26
Then T' = -0.04374*(84.26-24) = -2.6358
T = 87.36 - 2*2.6358 = 82.0884
Say 82.0884 after 4 minutes.
Do this three more times and you have an estimate of the temperature after 10 minutes.