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Certain quantity of water cools from $70^\circ C$ to $60^\circ C$ in the first 5 minutes and to $54^\circ C$ in the next 5 minutes. The temperature of the surroundings is:
A
$45^\circ C$
B
$20^\circ C$
C
$42^\circ C$
D
$10^\circ C$
Detailed Solution
Newton's law of cooling: $\frac{\theta_1 - \theta_2}{t} = k\left[\frac{\theta_1 + \theta_2}{2} - \theta_0\right]$
$\frac{70 - 60}{5} = k[65 - \theta_0] \Rightarrow 2 = k[65 - \theta_0]$ ...(i)
$\frac{60 - 54}{5} = k[57 - \theta_0] \Rightarrow \frac{6}{5} = k[57 - \theta_0]$ ...(ii)
Dividing (i) by (ii): $\frac{10}{6} = \frac{65 - \theta_0}{57 - \theta_0} \Rightarrow \theta_0 = 45^\circ C$
$\frac{70 - 60}{5} = k[65 - \theta_0] \Rightarrow 2 = k[65 - \theta_0]$ ...(i)
$\frac{60 - 54}{5} = k[57 - \theta_0] \Rightarrow \frac{6}{5} = k[57 - \theta_0]$ ...(ii)
Dividing (i) by (ii): $\frac{10}{6} = \frac{65 - \theta_0}{57 - \theta_0} \Rightarrow \theta_0 = 45^\circ C$