Two cups of tea at temperatures and cool in a room at ambient constant temperature . Initially .
Cup 1 has cool milk added instantaneously at and then hot water added at a constant rate after which is modelled as follows
whereas cup 2 is left undisturbed and evolves as follows
where and are the Dirac delta and Heaviside functions respectively, and is a positive constant.
(a) Derive expressions for when and for when .
(b) Show for that
(c) Derive an expression for for .
(d) At what time is ?
(e) Find how behaves for and explain your result.