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The temperature of an isolated black body falls from $T_{1}$ to $T_{2}$ in time t , let C be a constant, then

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$t = C [\frac{1}{{T_{2}}^{4}} - \frac{1}{{T_{1}}^{4}}]$

$t = C [\frac{1}{T_{2}} - \frac{1}{T_{1}}]$

$t = C [\frac{1}{{T_{2}}^{3}} - \frac{1}{{T_{1}}^{3}}]$

$t = C [\frac{1}{{T_{2}}^{2}} - \frac{1}{{T_{1}}^{2}}]$

answer is C.

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Detailed Solution

Power radiated $P = 4 π r^{2} σ T^{4}$
$- m s (\frac{d T}{d t}) = 4 π r^{2} σ T^{4}$
$\int_{T_{1}}^{T_{2}} \frac{1}{T^{4}} d T$ = constant $\int_{0}^{t} d T$
$C [\frac{1}{{T_{2}}^{3}} - \frac{1}{{T_{1}}^{3}}] = t [C \to c o n s \tan t]$

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