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24 November, 23:12

A thermometer initially reading 212F is placed in a room where the temperature is 70F. After 2 minutes the thermometer reads 125F. (a) What does the thermometer read after 4 minutes

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  1. 24 November, 23:24
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    91.3°F

    Explanation:

    Let T be the temperature of the thermometer at any time

    T∞ be the temperature of the room = 70°F

    T₀ be the initial temperature of the thermometer = 212°F

    And m, c, h are all constants from the cooling law relation

    From Newton's law of cooling

    Rate of Heat loss by the cake = Rate of Heat gain by the environment

    - mc (d/dt) (T - T∞) = h (T - T∞)

    (d/dt) (T - T∞) = dT/dt (Because T∞ is a constant)

    dT/dt = (-h/mc) (T - T∞)

    Let (h/mc) be k

    dT / (T - T∞) = - kdt

    Integrating the left hand side from T₀ to T and the right hand side from 0 to t

    In [ (T - T∞) / (T₀ - T∞) ] = - kt

    (T - T∞) / (T₀ - T∞) = e⁻ᵏᵗ

    (T - T∞) = (T₀ - T∞) e⁻ᵏᵗ

    Inserting the known variables

    (T - 70) = (212 - 70) e⁻ᵏᵗ

    (T - 70) = 142 e⁻ᵏᵗ

    At t = 2 minute, T = 125°F

    125 - 70 = 142 e⁻ᵏᵗ

    55/142 = e⁻ᵏᵗ

    - kt = In (55/142) = In (0.3873)

    - k (2) = - 0.9485

    k = 0.4742 / min

    At time t = 4 mins

    kt = 0.4742 * 4 = 1.897

    (T - 70) = 142 e⁻ᵏᵗ

    e^ (-1.897) = 0.15

    T - 70 = 142 * 0.15 = 21.3

    T = 91.3°F
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