Question

Newton's law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, dT K[M(t) - T(t)], where K is a constant. Let K=0.05 (min) - 1 and the temperature of the medium be constant, dt M(t) = 294 kelvins. If the body is initially at 370 kelvins, use Euler's method with h=0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes. (a) The temperature of the body after 30 minutes is kelvins. (Round to two decimal places as needed.)

Answer 1

Kindly go through the solution provided below.

Liven that dT = K k[M-T] dt where K = 6,05 and M= 294 We need to use Eulers method with step size (h=0.1) to approximate T (30) It is given that initially, the body temprature - 370 To = 370 And we know from the Eulers method that Yuri h dy ie f Yn = di So, we need to iterate from t=0 tot 30 with step size = 0.1 No.gl stefs= 30 = 300 => 0,1 Excel, which on a Here's is the iterations sniffet from were performed cell formula t Iteration number 283 284 285 286 287 288 289 290 D2 IX ✓ fx =$1$2*($I$3-C2) F dy at H A 1 Iteration number - h = K= M= с D E y(t) dydt y(t+h) 0 370 -3.8 369.62 0.1 369.62 -3.781 369.2419 0.2 369.2419 -3.7621 368.8657 0.3 368.8657 -3.74328 368.4914 0.4 368.4914 -3.72457 368.1189 0.1 0.05 294 2 3 4 5 6 3 y(t) dydt y(t+h) 28.2 312.4894 -0.92447 312.3969 28.3 312.3969 -0.91985 312.3049 28.4 312.3049 -0.91525 312.2134 28.5 312.2134 -0.91067 312.1224 28.6 312.1224 -0.90612 312.0317 28.7 312.0317 -0.90159 311.9416 28.8 311.9416 -0.89708 311.8519 28.9 311.8519 -0.89259 311.7626 29 311.7626 -0.88813 311.6738 29.1 311.6738 -0.88369 311.5854 29.2 311.5854 -0.87927 311.4975 29.3 311.4975 -0.87488 311.41 29.4 311.41 -0.8705 311.323 29.5 311.323 -0.86615 311.2364 29.6 311.2364 -0.86182 311.1502 29.7 311.1502 -0.85751 311.0644 29.8 311.0644 -0.85322 310.9791 29.9 310.9791 -0.84896 310.8942 30 310.8942 -0.84471 310.8097 4 5 E2 X fic 291 292 293 294 295 296 297 298 299 300 301 =C2+$I$1*D2 F H t 2 3 Iteration number 1 2 3 4 с y(t) dydt ytth) 0 370 -3.8 369.62 0.1 369.62 -3.781 369.2419 0.2 369.2419 -3.7621 368.8657 0.3 368.8657 -3.74328 368.4914 h= K= M= 0.1 0.05 294 5 cell formule for calculating Ynt This is the nt= y (#+) result y (30) Corresponding to t= 30