A hot body is being cooled in air according to Newton's law...

A hot body is being cooled in air according to Newton's law of cooling, the rate of fall of temperature being $k$ times the difference of its temperature with respect to that of surroundings. The time, after which the body will lose half the maximum heat it can lose, is (A) $\frac{1}{k}$ (B) $\frac{\ln 2}{k}$ (C) $\frac{\ln 3}{k}$ (D) $\frac{2}{k}$

A hot body is being cooled in air according to Newton's law of cooling, the rate of fall of temperature being $k$ times the difference of its temperature with respect to that of surroundings. The time, after which the body will lose half the maximum heat it can lose, is
(A) $\frac{1}{k}$
(B) $\frac{\ln 2}{k}$
(C) $\frac{\ln 3}{k}$
(D) $\frac{2}{k}$

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Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling states that the rate at which an object cools (or heats) is proportional to the difference between its temperature and the ambient temperature. This principle is typically modeled by a differential equation where the rate of temperature change is proportional to the difference between the object's temperature and that of its surroundings.

Exponential Decay

Exponential decay refers to the decrease of a quantity at a rate proportional to its current value. In the context of cooling, the temperature difference between an object and its environment decreases exponentially over time. This is characterized by a decay constant that governs the speed of the process.

Half-life

The half-life in an exponential decay process is the amount of time required for the quantity of interest (in this case, the temperature difference or the maximum heat content) to decrease by half its initial value. It is determined using the decay constant, and mathematically, the half-life time can be found by setting the exponential decay equation equal to one-half and solving for time.

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