Question

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

Answer 1

$\huge{ \underline{ \tt \red{Answer - }}}$

Given - The temperature of a body falls from 40°C to 36°C in 5 minutes when placed in a surrounding of constant temperature 16°C.

To find :- Time taken for the temperature of the body to become 32°C.

$\large{ \underline \bold{ \tt{ \blue{Solution:-</p><p>}}}}$

Mean temperature of the body as it cools from 40°C to 36°C

$\large{ \rm{ = \frac{40°C - 36°C}{2} = 38°C}}$

The rate of decrease of temperature

$\rm{ = \frac{40°C - 36°C}{5 \: min} = 0.80°C/min}$

Now, according to newton's law of cooling,

$\large{ \red{\frac{d \theta}{dt} = - k( \theta \: - \theta_0)}}$

$\implies \: - 0.8°C/min = - k(38°C - 16°C)$

$\implies \: k = \frac{0.8}{22}$

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

During this period,

$\implies \: \frac{d \theta}{dt} = - \frac{36°C - 32°C }{t} = - \frac{4°C }{t}$

The mean temperature

$= \frac{36°C + 32°C }{2} = 34°C$

$\large{ \red{\frac{d \theta}{dt} = - k( \theta \: - \theta_0)}}$

$= - \frac{4°C }{t} = - \frac{0.8}{22} \times (34°C - 16°C )/min$

$\boxed{ \rm{ \red{\therefore \: t = 6.1 \: min}}}$

Hence Time taken for the temperature of the body to become 32°C is 6.1 min