1) A thermometer with an initial temperature of 75°C is placed in a room with a controlled temperature of 22°C. After 3 minutes, the thermometer reads 60°C, determine how long will the thermometer reads 30°C. Determine also the reading after 8 minutes.
1) A thermometer with an initial temperature of 75°C is placed in a room with a controlled temperature of 22°C. After 3 minutes, the thermometer reads 60°C, determine how long will the thermometer reads 30°C. Determine also the reading after 8 minutes.
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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Transcribed Image Text:1) A thermometer with an initial temperature of 75°C is placed in a room with a controlled
temperature of 22°C. After 3 minutes, the thermometer reads 60°C, determine how long will the
thermometer reads 30°C. Determine also the reading after 8 minutes.
Transcribed Image Text:Newton's Law of Cooling
Newton's Law of Cooling states that the time rate of change in the temperature of the body
is directly proportional to the temperature difference between the body and its surrounding medium
where the temperature is held constant.
Let u be the temperature of the body and T is the temperature of the surrounding medium
where the body is immersed into then by definition of the Newton's law of cooling:
du
dt
: = k(T – u)
or can be written as
du
dt
-k(u – T)
where k > 0 and is called the thermometer constant
Finding the solution to this equation
du
-k(и — Т)
dt
du
(-k) dt
и — Т
In(u – T) = -kt + c
и — Т%3D се-kt
then
u = T + ce-kt
Another form of this equation from other references is
T = T, + (T, – T,)e-kt
where T – temperature of the body at any time, t
T; – surrounding or ambient temperature
To – initial temperature of the body, or temperature at t = 0
k – thermometer constant; k > 0
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