A differential equation for the temperature ' $\mathrm{T}$ ' of a hot body as a function of time, when it is…

A differential equation for the temperature ' $\mathrm{T}$ ' of a hot body as a function of time, when it is placed in a both which is held at a constant temperature of $32^{\circ} \mathrm{F}$, is given by (where $\mathrm{k}$ is a constant of proportionality)
  1. $\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{kT}-32$
  2. $\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{kT}+32$
  3. $\frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-32)$
  4. $\frac{\mathrm{dT}}{\mathrm{dt}}=32 \mathrm{kT}$

Solution

The temperature $\mathrm{T}$ of the body will decrease with time. The body is kept in a bath of temperature $32^{\circ} \mathrm{F}$. $\begin{aligned} & \therefore \frac{\mathrm{dT}}{\mathrm{dt}} \alpha-(\mathrm{T}-32) \\ & \Rightarrow \frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-32) \end{aligned}$

Asked in: MHT CET 2021 (20 Sep Shift 1)

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