Question

the rate of cooling of water is .... when the different in the temperature of water and the ambience is large

Answer 1

When an object is at a different temperature than its surroundings, it will gradually cool down or heat up until the temperatures are equal.  Everyone has experienced this.  You boil water to make tea and then wait several minutes until it is at a temperature at which you can drink it.  You place a cold turkey in the hot oven for Thanksgiving dinner and after several hours it has reached the desired temperature.  Newton’s Law of Cooling relates the rate of change in the temperature to the difference in temperature between an object and its surroundings.

                       dT/dt = k(T - Tα)

where T =  the temperature of the object of interest as a function of time

            t =  the time

               Tα = the ambient temperature

                    T∞ = the initial temperature

             k = a proportionality constant specific to the object of interest

This differential equation can be integrated to produce the following equation.

        T(t) = tα + (t∞ - tα)$e^{kt}$