Modeling equations help

Question

A car engine running at a temperature of 80◦C is then switched off and begins to cool according to Newton’s Law of Cooling. The ambient temperature is 15◦C, and after exactly 18 minutes and 9 seconds the engine temperature has reduced to 50◦C.
(a) Write down a model for the cooling of the engine using Newton’s Law of Cooling in the form T(t) = Ts + (T0 − Ts)e−kt 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.
(b) Hence determine the rate constant, k.
(c) How long after the engine was turned off does the engine temperature drop to 30◦C?

Answer 1

I'll do the first part (b); substitution gives  50 = 15 + (80-15)e^-18.15k because 18 min 9 sec = 18.15 min.

35 = 65 e^-18.15k

0.543846 = e^-18.15k  and now we take natural logarithms to get

-0.6190392  = -18.15k, so

k = 0.034106843.  (I believe in using many decimal places, to minimize errors due to roundoff in forthcoming calculations; you may round the final answer when you do part (c)