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18 September, 19:25

A thermometer is taken from a room where the temperature is 24oc to the outdoors, where the temperature is - 15oc. After one minute the thermometer reads 14oc. (a) what will the reading on the thermometer be after 3 more minutes?

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  1. 18 September, 19:53
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    Solution:

    Use Newton's Law of Cooling.

    T = T_s + (T_0 - T_s) * e^ (-kt)

    where

    T = temperature at any instant

    T_s = temperature of surroundings

    T_0 = original temperature

    t = elapsed time

    k = constant

    Now, we need to find this constant. We are given that after one hour, the temperature drops to 13° C in a 7°C Environment.

    T = 14, T_0 = 24, T_s = - 15, t = 1, k = ?

    T = T_s + (T_0 - T_s) * e^ (-kt)

    ==> 14 = - 15 + (24 - 7) * e^ (-k)

    ==> 14 = 7 + 17*e^ (-k)

    ==> 7 = 17*e^ (-k)

    ==> 7/13 = e^ (-k)

    ==> - k = ln (7/17)

    ==> k = - ln (7/17) ≈ 0.774

    Now,

    Let's calculate temperatures!

    T = ?, T_0 = 24, T_s = - 15, k = 0.773, t = 3

    T = T_s + (T_0 - T_s) * e^ (-kt)

    ==> T = - 15 + (24 - (-15)) * e^[ - (0.774) (2) ]

    ==> T = - 15 + 39*e^ (-1.548)

    ==> T ≈ 15.72° C

    This the required answer.
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