ton'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, = K[M(t) - T(t)), where K is a stant. Let K= 0.03 (min) and the temperature of the medium be constant, M(t) = 290 kelvins. If the body is initially at 358 kelvins, use Euler's method with h = 0.1 min to approximate the temperature of the body after (a) 30 minutes and minutes. The temperature of the body after 30 minutes is und to two decimal places as needed.) kelvins.

ton'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, = K[M(t) - T(t)), where K is a stant. Let K= 0.03 (min) and the temperature of the medium be constant, M(t) = 290 kelvins. If the body is initially at 358 kelvins, use Euler's method with h = 0.1 min to approximate the temperature of the body after (a) 30 minutes and minutes. The temperature of the body after 30 minutes is und to two decimal places as needed.) kelvins.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

dT
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,
= K[M(t) – T(t)], where K is a
dt
- 1
constant. Let K = 0.03 (min)' and the temperature of the medium be constant, M(t) = 290 kelvins. If the body is initially at 358 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.)

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According to Newton's law of cooling, the temperature of a body changes at a rate proportional to the difference between the temperature of the body and the temperature of the surrounding medium. Thus, if Tm is the temperature of the medium and T = T(t) is the temperature of the body at time t, then T'= -k(T - Tm) where k is a positive constant. In this exercise, you will verify that T = Tm+ (To-Tm)e -kt is a solution to the differential equation. Here To denotes the initial temperature T(0) of the body. (a) First substitute the above expression for T into the left side of the differential equation. In other words, compute T'. Use an underscore "_" to write the subscripts on To and Im T'= (b) Next substitute the above expression for T into the right side of the differential equation. In other words, compute -k(T – Tm). - -k(T - Tm) (c) Are your answers in parts (a) and (b) equal? No Yes