The temperature T of a cooling object drops at a rate proportional to the difference `T - S` where S is constant temperature of surrounding medium. If initially `T = 150^@C`; find the temperature of the cooling object at any time t.

Explanation

Let `{T}` be the temperature of the cooling object at any time`{t}`.


Given, `frac{{dT}}{{dt}} propto({T}-{S})`


` Rightarrow frac{{dT}}{{dt}}=-{K}({T}-{S})` where `{k}` is negative


`Rightarrow frac{{dT}}{{T}-{S}}=-{k} cdot {dt}`


`Rightarrow log ({T}-{S})=-{kt}+log {c}`


`Rightarrow log ({T}-{S})=-log {c}={kt}`


`Rightarrow log frac{{T}-{S}}{{c}}=-{kt}`


` Rightarrow log frac{{T}-{S}}{{c}}={e}^{-{kt}}`


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