QUESTION 1 (Total marks for Question 1: 20) (a) Hot water at a constant rate of 2 1/min and temperature Th(t) is mixed with cold water at a constant rate of 3 l/min and a constant temperature of 20C. Both streams flow into a bathtub, but because of carelessness, the water is over-flowing and keeping the bathtub full of water. The volume of the bathtub is 100 1. Assuming the water in the bathtub is perfectly mixed, derive the differential equation relating the temperature in the bathtub, T(t), to the temperature of the hot water, Th(t). Obtain the transfer function G(s)=T(s)/Th(s) and state its gain and time constant. (b) Draw the block diagram representing each of the transfer functions i-iii below, where Xi(s) represents the i-th input and Y(s) represents the i-th output. In each case, do not perform any algebraic manipulations to simplify the transfer functions, but use the rules of block diagram algebra to simplify the diagram if possible. i. Y(s)= =- K TS+1 K, -X (s) + -X(s) TS+1 1 ii. Y,(s)-(K,X,(s)-K.X,(s)) TS+1 Y(s)=G(s)X(s)+G(s) Y(s) iii. Y(s)=G(s)x, (s)