Question

The temperature of a body in air falls from 40°C to 24°C in 4 minutes. The temperature of the air is 16°C. The temperature of the body in the next 4 minutes will be

• A. 16°C
• B. 20°C
• C. 56/3°C
• D. 22°C

Answer 1

Answer: (C)

56/3°C

Answer 2

Newton's Law of Cooling states that for equal time intervals, the ratio of successive temperature differences (body temperature minus surroundings temperature) is constant.

Initial temperature difference: $T_1 - T_s = 40^\circ\text{C} - 16^\circ\text{C} = 24^\circ\text{C}$. Temperature difference after 4 minutes: $T_2 - T_s = 24^\circ\text{C} - 16^\circ\text{C} = 8^\circ\text{C}$.

The ratio of temperature differences is $\frac{8}{24} = \frac{1}{3}$.

Let $T'_2$ be the temperature after the next 4 minutes. The temperature difference at the start of this interval is $8^\circ\text{C}$. Using the same ratio: $\frac{T'_2 - T_s}{T_2 - T_s} = \frac{1}{3}$. $\frac{T'_2 - 16}{8} = \frac{1}{3}$. $T'_2 - 16 = \frac{8}{3}$. $T'_2 = 16 + \frac{8}{3} = \frac{48+8}{3} = \frac{56}{3}^\circ\text{C}$.