A thermometer is taken from a room where the temperature is 21 degrees to the ou

Question

A thermometer is taken from a room where the temperature is 21 degrees to the outdoors, where the temperature is 2degrees. After one minute the thermometer reads 14degrees. Remember that Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings.
(a) What will the reading on the thermometer be after 5 more minutes?
,
(b) When will the thermometer read 3degrees?
minutes after it was taken to the outdoors.

Explanation / Answer

a) from newtons law of cooling

T(t) = Ts + C e^kt

Ts = 2 degrees

T(0) = 21

21 = 2 + C

C = 19

and T(1) = 14

2 + 19 e^k = 14

k = ln(12/19)

T(t) = 2 + 19 * (12/19)^t

a)

T(5) = 2 + 19 * (12/19)^5) = 3.909 degrees

b) 3 = 2 + 19 * (12/19)^t

t = 6.408 s

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