A body cools according to Newton's law of cooling from 100^ C to 60^ C in 20 minutes. The temperature of the surroundings being 20^ C, then the total time required for the body to cool down to 30^ C is

Step-by-step solution

According to Newton's law of cooling, the temperature of a body at time t is given by: T - T_s = (T₀ - T_s) e^ -kt Given T₀ = 100^ C and T_s = 20^ C. At t = 20 minutes, T = 60^ C. Substituting these values into the equation: 60 - 20 = (100 - 20) e^ -20k 40 = 80 e^ -20k e^ -20k = 1 2 Taking the natural logarithm on both sides: -20k = - 2 k = 2 20 Let t be the total time required for the body to cool down to 30^ C. 30 - 20 = (100 - 20) e^ -kt 10 = 80 e^ -kt e^ -kt = 1 8 = ( 1 2 )^3 Taking the natural logarithm again: