A hot body placed in a surrounding of temperature ₀ conforms to Newton's law of cooling, d dt = -k( - ₀) . At t = 0 , its temperature is ₁ . The mass of the body is m and its specific heat capacity is s . Determine the time, starting from t = 0 , in which the body will lose 90 % of the maximum heat it can possibly lose.

Step-by-step solution

The maximum heat the body can lose is when its temperature drops from ₁ to the surrounding temperature ₀ . Maximum heat loss, Q_ max = ms( ₁ - ₀) The heat lost by the body at time t when its temperature is is given by: Q = ms( ₁ - ) Given that the body loses 90 % of the maximum heat it can possibly lose: Q = 0.9 Q_ max ms( ₁ - ) = 0.9 ms( ₁ - ₀) ₁ - = 0.9( ₁ - ₀) - ₀ = 0.1( ₁ - ₀) From Newton's law of cooling, the temperature at time t is given by: - ₀ = ( ₁ - ₀)e^ -kt Substituting the value of - ₀ : 0.1( ₁ - ₀) =