Answer to Question #346468 in Differential Equations for sami

Question #346468

A thermometer is removed from a room where the temperature is 70° F and is taken outside, where the air temperature is 10° F. After one-half minute the thermometer reads 50° F. What is the reading of the thermometer at t  1 min? How long will it take for the thermometer to reach 15° F?


1
Expert's answer
2022-05-31T13:55:11-0400

Let "T="the temperature (in "\\degree F" ) recorded by the thermometer at time "t."

The equation governing "T" is obtained from Newton’s Law of Cooling:


"\\dfrac{dT}{dt}=k(T-10)"

where "k" is some constant.

We are given that "T(0) = 70" and "T( 1\/ 2 ) = 50."


"\\dfrac{dT}{T-10}=kdt"

Intergrate


"\\int \\dfrac{dT}{T-10}=\\int kdt"

"\\ln(T-10)=kt+\\ln C"

"T-10=Ce^{kt}"

"T=Ce^{kt}+10"

"T(0)=C+10=70=>C=60"

"T(1\/2)=60e^{k(1\/2)}+10=50"

"e^{k(1\/2)}=2\/3"


"k=2\\ln(\\dfrac{2}{3})"

"T(t)=60(\\dfrac{4}{9})^t+10"

a)


"T(1)=60(\\dfrac{4}{9})^1+10=\\dfrac{110}{3}(\\degree F)\\approx36.67(\\degree F)"

b)


"60(\\dfrac{4}{9})^t+10=15"

"(\\dfrac{4}{9})^t=\\dfrac{1}{12}"

"t=\\dfrac{\\ln(1\/12)}{\\ln(4\/9)}\\ min\\approx3.06\\ min"


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