Question

A differential equation for the temperature 'T' of a hot body as a function of time, when it is placed in a bath which is held at a constant temperature of $32^\circ\text{F}$, is given by (where k is a constant of proportionality)

• A. $\frac{\text{d}T}{\text{d}t} = k(T - 32)$
• B. $\frac{\text{d}T}{\text{d}t} = kT + 32$
• C. $\frac{\text{d}T}{\text{d}t} = -k(T - 32)$
• D. $\frac{\text{d}T}{\text{d}t} = 32kT$

Hint:

Always look for the negative sign when a hot object is cooling down! Since the temperature must decrease over time, $\frac{\text{d}T}{\text{d}t}$ must be negative, which immediately points you to option (C).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to formulate the differential equation describing the rate of change of temperature of a hot body over time when placed in a constant-temperature surrounding bath using Newton's Law of Cooling.

Step 2: Detailed Explanation:
According to Newton's Law of Cooling, the rate of change of temperature ($\frac{\text{d}T}{\text{d}t}$) of an object is directly proportional to the difference between its own temperature ($T$) and the constant ambient temperature of the surroundings ($T_s$): $$ \frac{\text{d}T}{\text{d}t} \propto (T - T_s) $$ Here, the surrounding bath temperature is given as $T_s = 32^\circ\text{F}$. Since the body is hot, its temperature will decrease over time, meaning the rate of change is negative: $$ \frac{\text{d}T}{\text{d}t} = -k(T - 32) $$ where $k$ is a positive constant of proportionality.

Step 3: Final Answer:
The correct differential equation is represented by option (C).