Question

A body initially at 80 ${}^\circ C$ cools to 64${}^\circ C$ in 5 min and to 52 ${}^\circ C$ in 10 min. the temperature of the surrounding is?
A. ${{26}^{0}}C$
B. ${{16}^{0}}C$
C. ${{36}^{0}}C$
D. ${{40}^{0}}C$

Answer 1

This is a problem of Newton’s law of cooling as we have to determine the temperature of the surrounding. We are given two conditions in which a body cools down from a certain temperature to another, we can make use of Newton’s law of cooling to find two relationships and then we can find the value of constant k and find the temperature of the surrounding. We have two unknowns and so, we need two equations. Remember the rule, the number of unknown variables, the same number of equations we need to solve the same.

Complete step by step answer:
According to Newton's Law of Cooling, rate of heat transfer is given by
$\dfrac{{{T}_{2}}-{{T}_{1}}}{t}=k(\dfrac{{{T}_{2}}+{{T}_{1}}}{2}-{{T}_{0}})$
For the first case:
$\Rightarrow \dfrac{80-64}{5}=k(\dfrac{80+64}{2}-{{T}_{0}})$
$\Rightarrow 3.2=k(72-{{T}_{0}})$----(1)
For the second case:
$\Rightarrow \dfrac{64-52}{5}=k(\dfrac{64+52}{2}-{{T}_{0}})$
$\Rightarrow 2.4=k(58-{{T}_{0}})$---(2)
Dividing the (1) equation by (2) we get,
$185.6-3.2{{T}_{0}}=172.8-2.4{{T}_{0}}$
$\therefore {{T}_{0}}={{16}^{0}}C$

So, the correct option is B.

Note: Newton’s law of cooling tells us the rate at which a body exposed to atmosphere changes temperature through by loss of heat to the surrounding which is approximately proportional to the difference between the object’s temperature and its surroundings, provided the difference is small. As per this law rate of cooling is dependent upon the temperature, suppose we want a hot cup of tea to cool down, a cup of hot coffee will cool more quickly if we put it in the refrigerator.