A body at 50 degrees cools in surrounding maintained at 30 degrees the temperature at which the rate of cooling is half that of the beginning

We will solve this problem using Newton’s Law of Cooling, which states:

Rate of cooling = k (T - T_s)

where

T is the temperature of the body at any time t,
T_s is the surrounding temperature,
k is a constant.
Step 1: Define Initial Conditions
Initial temperature of the body, T_0 = 50°C
Surrounding temperature, T_s = 30°C
Initial rate of cooling:
(dT/dt)₀ = k (50 - 30) = 20k
Step 2: Find Temperature When Rate of Cooling is Half
We need to find the temperature T at which the rate of cooling is half of the initial rate.

So,
(dT/dt) = (1/2) * (dT/dt)₀
k (T - 30) = (1/2) * (20k)
T - 30 = (1/2) * 20
T - 30 = 10
T = 40°C

Final Answer:
The temperature at which the rate of cooling is half of the initial rate is 40°C.