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# If a piece of metal is heated to temperature $\theta$ and then allowed to cool in a room which is at temperature, the graph between the temperature ${{\theta }_{0}}$, the graph between the temperature of the metal and time t will be closest to :(A)(B)(C )(D)

Last updated date: 10th Aug 2024
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Hint: Apply Newton’s law of cooling. According to Newton’s law of cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. By applying this concept and doing integration by parts we will get the solution and from that solution we can find the type of graph used to plot them.
Formula used:
$\dfrac{dT}{dt}=k({{T}_{2}}-{{T}_{1}})$

$\dfrac{dT}{dt}=k({{T}_{2}}-{{T}_{1}})$
Rearranging them we get,
$\dfrac{dT}{{{T}_{2}}-{{T}_{1}}}=kdt$
Integrating on both sides we get,
$\log ({{T}_{2}}-{{T}_{1}})=-kt$
$\Rightarrow$ ${{T}_{2}}-{{T}_{1}}={{e}^{-kt}}$
$\Rightarrow$ ${{T}_{2}}={{T}_{1}}+{{e}^{-kt}}$
According to Newton’s law of cooling, the temperature goes on decreasing with time exponentially. Here we can see that temperature is exponentially decreasing. Then the graph is,

Therefore option(B) is correct.