Question

Part B: Answer all questions by showing your solving step by step. 34 marks) 9. A cup of tea is prepared at 190F in a room with temperature 80F. After 3 minutes the tea has cooled to 175F [10marks) a) Find an equation of the temperature at any time. b) Find the time t at which the tea has cooled to 130F 10. The half-life time of Morphine in human body is 3.5 hours. If initially there is 0.5 mg of [8marks) Morphine in the blood. a) b) Find the amount of Morphine in the body at any time Determine when the amount drops to 0.02 mg.

Answer 1

Equation of temperature T() is T(t) = T_a + (T_0 - T_1) e^- k t T_a = ambecent temperature T_0 = initial temperature k = proportionality constant T_0 = 104 F T_a = 80 F T(t) = 175 F (at t = 3) So 175 = 80 + (104 - 80) e^- k(3) 175 - 80 = (104 - 80) e^- 3 k 95 = 24 e^- 3 k 95/24 = e^- 3 k 3.9583 = e^- 3 k Rightarrow ln(3.9583) = ln(e^- 3 k) 1.3758 = -3 k k = -1.3758/3 k = -0.4586 So, T(t) = 80 + 24 e^0.45867 at T(t) = 13 F So, 130F = 80 + 24 e^0.4586 t 50 = 24 e^0.4586 t 50/24 = e^0.4586 t Rightarrow 2.083 = e^0.4586 t Rightarrow ln(2.083) = 0.4586 t Rightarrow 0.7228 = 0.4586 t Rightarrow 1.60008 = t Rightarrow t = 1.6 min.