A liquid is heated so that its temperature is x°C after t seconds. It is given that the rate of increase of x is proportional to (100 − x). 

The initial temperature of the liquid is 25°C.
a. Form a differential equation relating x, t and a constant of proportionality, k, to model this information.

b. Solve the differential equation and obtain an expression for x in terms of t and k.

c. After 180 seconds the temperature of the liquid is 85°C. Find the temperature of the liquid after 195 seconds.

d. The model predicts that x cannot exceed a certain temperature. Write down this maximum temperature.