Correct Answer - Option 3 : 7 min
CONCEPT:
Newton’s law of cooling
- According to Newton’s law of cooling, the rate of loss of heat from a body is directly proportional to the difference in the temperature of the body, and its surroundings.
- Mathematically it is given as,
\(⇒ \frac{dT}{dt}=k(T_{t}-T_{s})\)
Where
- Tt = temperature at time t
- Ts = temperature of the surrounding
- k = Positive constant that depends on the area and nature of the surface of the body under consideration
- dt = time required
CALCULATION:
For case 1:
⇒ Ts = 20°C
⇒ dT = 70 - 50
⇒ dT = 20°C
⇒ dt = 7 min
\(⇒ T_{t}=\frac{70+50}{2}\)
⇒ Tt = 60°C
\(⇒ \frac{dT}{dt}=k(T_{t}-T_{s})\)
\(⇒ \frac{20}{7}=k(60-20)\)
\(⇒ \frac{20}{7}=40k\) -----(1)
For case 1:
⇒ Ts = 20°C
⇒ dT = 50 - 38
⇒ dT = 12°C
\(⇒ T_{t}=\frac{50+38}{2}\)
⇒ Tt = 44°C
\(⇒ \frac{dT}{dt}=k(T_{t}-T_{s})\)
\(⇒ \frac{12}{dt}=k(44-20)\)
\(⇒ \frac{12}{dt}=24k\) -----(2)
By equation 1 and equation 2,
⇒ dt = 7 min
- Hence, option 3 is correct.