A body is kept in cooler surroundings. When the temperature of the body is $70^{0} C$ , it cools at the rate of $3^{0} C$ per minute and at $50^{0} C$ , it cools at the rate of ${1.5}^{0} C$ per minute. Find the temperature of the body when it cools at the rate of $1^{0} C$ per minute.

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Solution

Rate of cooling $\frac{d T}{d t} = \frac{- k}{m s} (T - T_{s})$

Or $3 = \frac{- k}{m s} (70 - T_{s})$ .....(1)

Or $1.5 = \frac{- k}{m s} (50 - T_{s})$ ......(2)

So, $\frac{3}{1.5} = \frac{70 - T_{s}}{50 - T_{s}}$ $⟹ T_{s} = 30^{o} C$

When rate of cooling is $1^{o} C$ per minute
So, $1 = \frac{- k}{m s} (T - 30)$ .....(3)

Equation (3)/(1), $\frac{1}{3} = \frac{T - 30}{70 - 30}$

$⟹ T = {43.33}^{o} C$

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