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The temperature of a body falls from 40C to 36C in 5 minutes when placed in a surrounding of constant temperature 16C. Find the time taken for the temperature of the body to become 32C.


  1. 8.1 min

  2. 6.1 min

  3. 5.1 min

  4. 5.3 min


Solution

The correct option is B

6.1 min


As the temperature differences are small, we can use Newton's law of cooling.

dθdt=kt(θθ0)

dθθθ0=kdt

or,                                               ...(i)

Where k is a constant, θ is the temperature of the body at time t and θ0=16C is the temperature of the surrounding. We have,

36C40Cdθθθ0=kt(5 min)

ln36C16C40C16C=kt(5 min)

or,

                                     k=ln(56)5 min

or, .

If t be the time required for the temperature to fall from 36C to 32C then by (i),

32C36Cdθθθ0=kt

or,        ln32C16C36C16C=ln(56)5 min

or,          t=ln(45)(ln(56)×5min

= 6.1 min.

Alternative method

The mean temperature of the body as it cools from 40C to 36C is  40C+36C2=38C The rate of decrease of temperature is

  40C+36C5min=0.80C/min .

Newton's law of cooling is

dθdt=k(θθ0)

or, -0.8C/min = - k(38C - 16C)=-k(22C)

or, k=0.822min1.

Let the time taken for the temperature to become 32C be t.

During this period,.

The mean temperature is

Now,

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