Question

In a room where the temperature is 30°C a body cools from 61°C to 59°C in 4 minutes. The time taken by the body to cool from 51°C to 49°C will be:     
1. 4 min
2. 5 min
3. 6 min
4. 8 min

Answer 1

Correct Answer - Option 3 : 6 min

CONCEPT:

• Newton's law of cooling: It describes the rate at which an exposed hot body cools to the cooler temperature of the environment through radiation.

The cooling rate is calculated by 

\( {dT \over dt}=-k(T-T_0)\)

where \( {dT \over dt}\) is the rate of cooling, T is the temperature of the body at time T, T0 is the temperature of the environment, and k is the constant.

CALCULATION:

From Newton's Law of cooling \( {dT \over dt}=-k(T-T_0)\)

\( {dT \over (T-T_0)}=-k dt\)

integrating both sides with limits

\(ln {T_2-T0 \over (T_1-T_0)}=-kt\)

\(\frac{1}{t}ln {T_2-T0 \over (T_1-T_0)}=contant\)

\(\frac{1}{4}ln {(59-30) \over (61-30)}=\frac{1}{t}ln {(49-30) \over (51-30)}\)

t = 6 min

So the correct answer is option 3.