Question

(30 pts) Newton's law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature). dT * = -k(T – Ta) where T = the temperature of the body (°C), t = time (min), k = the proportionality constant (per minute), and Ta = the ambient temperature (°C). Suppose that a cup of coffee originally has a temperature of 80 °C. Use Euler's method to manually (not using MATLAB) compute the temperature from t = 0 to 4 min using a step size of 2 min if Ta = 25°C and k = 0.021/min.

Answer 1

Given data →

k=0.021 /min

Ambient temperature (T) = 25 °C

Step size (h) = 2 min

Initial temperature (T;) = 80 °C

di = -0.021 * (T; – 25)

Time in min (t)	Initial temperature (T;)	di = -0.021 * (T; – 25)	new Ti = old T; +h*
0 min	0, 08	dT = -0.021 * (80 – 25) = -1.155	new T; = 80 + 2*(-1.155) = 77.69 °C
2 min	77,69 'C	-=-0.021 + (77.69-25) = -1.10649	new T; = 77.69 +2*(-1.10649) = 75.47702°C
4 min	75.47702 C

Summary of result

Time in min (t)	0.00	2.00	4.00
Temperature in °C	0, 08	77,69 'C	75.47702 C