If a metallic sphere gets cooled from $62^\circ \mathrm{C}$ to $50^\circ \mathrm{C}$ in 40° C and in the next 10 minutes gets cooled to 42^\circ C , then the temperature of the surroundings is
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$\frac{\theta_1 - \theta_2}{t} = K \left[ \frac{\theta_1 + \theta_2}{2} - \theta_0 \right]$
In the first 10 minute
$\frac{62-50}{10} = K \left[ \frac{62+50}{2} - \theta_0 \right]$ ⇒ ⇒ $1.2 = K[56 - \theta_0]$ .... (i)
$\frac{50-42}{10} = K \left[ \frac{50+42}{2} - \theta_0 \right]$ ⇒ ⇒ $0.8 = K[46 - \theta_0]$ .... (ii)
from equations (i) and (ii)
$\frac{1.2}{0.8} = \frac{(56 - \theta_o)}{(46 - \theta_o)}$ ⇒ ⇒ $\theta_0 = 26^{\circ}C$
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