Category : JEE Main & Advanced
(1) Curve between \[log(\theta -{{\theta }_{0}})\] and time
As \[\frac{d\theta }{dt}\propto -(\theta -{{\theta }_{0}})\Rightarrow \,\frac{d\theta }{(\theta -{{\theta }_{0}})}=-Kdt\]
Integrating \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+C\] \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+{{\log }_{e}}A\]
This is a straight line with negative slope
(2) Curve between temperature of body and time
As \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+{{\log }_{e}}A\]\[\Rightarrow \]\[{{\log }_{e}}\frac{\theta -{{\theta }_{0}}}{A}=-Kt\]
\[\Rightarrow \] \[\theta -{{\theta }_{0}}=A{{e}^{-kt}}\]
which indicates temperature decreases exponentially with increasing time.
(3) Curve between the rate of cooling (R) and body temperature \[(\theta )\].
\[R=K(\theta -{{\theta }_{0}})=K\theta -K{{\theta }_{0}}\] This is a straight line intercept R-axis at \[-K{{\theta }_{0}}\]
(4) Curve between rate of cooling (R) and temperature difference between body \[(\theta )\] and surrounding \[({{\theta }_{0}})\]
\[R\propto (\theta -{{\theta }_{0}})\]. This is a straight line passing through origin.
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