(Solved): (25 points) A thermometer that can be represented by a first-order transfer function having a time ...

(25 points) A thermometer that can be represented by a first-order transfer function having a time constant of \( 0.2 \mathrm{~min} \) is placed in a temperature bath, and after the thermometer comes to equilibrium with the bath, the temperature of the bath is increased linearly with time at a rate of \( 1 \% \) min. Find the difference between the indicated temperature and the bath temperature a) \( 0.1 \) min after the change in temperature begins, and b) \( 1.0 \) min after the change in temperature begins. c) What is the maximum deviation between indicated temperature and bath temperature, and when does it occur? d) Sketch the forcing (input) function and response on the same graph. The sketch does not need to be to scale. e) After a long time, by how many minutes does the response lag the input?