A car engine running at a temperature of 80C is then switched off and begins to cool according to Newton's Law of Cooling. The ambient temperature is 15C, and after exactly 18 minutes and 9 seconds the engine temperature has reduced to 50C. Write down a model for the cooling of the engine using Newton's Law of Cooling in the form T(t) = Ts + (T0 Ts)ekt where t is time measured in minutes, T0 is the initial temperature of the engine (in degrees Celsius), Ts is the ambient temperature (in degrees Celsius), T(t) is the engine temperature (in degrees Celsius) at time t and k is a rate constant. Here constructing the model involves substituting the initial engine temperature and the ambient temperature into the given equation for Newton's Law of Cooling, and then simplifying the equation where possible.

Question 1. (4 marks) A car engine running at a temperature of 80C is then switched off and begins to cool according to Newton's Law of Cooling. The ambient temperature is 15C, and after exactly 18 minutes and 9 seconds the engine temperature has reduced to 50C. (a) W'rite down a model for the cooling of the engine using Newton's Lats.r of Cooling in the form T{t) = T,5 + (To Ts)e'kt where t is time measured in minutes, To is the initial temperature of the engine (in degrees Celsius), T5 is the ambient temperature (in degrees Celsius}, T(t) is the engine temperature (in degrees Celsius) at time t and k is a rate constant. Here constructing the model involves substituting the initial engine temperature and the ambient temperature into the given equation for Newton's Law of Cooling: and then simplifying the equation where possible. (1)) Hence determine the rate constant, k. (c) How long after the engine was turned off does the engine temperature drop to 30C