a$).$ A mercury thermometer having a time constant of $0\mathrm{.}1$ min is placed in a temperature bath at $100\backslash$deg F and allowed to come to equilibrium with the bath. At time t$=0,$ the temperature of the bath begins to vary sinusoidally about

its average temperature of $100$F with an amplitude of

$2$F$.$ If the frequency of oscillation is $10/-$

cycles$/$min$,$

plot the ultimate response of the thermometer reading

as a function of time. What is the phase lag?

$($b$)$ A tank having a time constant of $1$ min and a resistance

of $1/9$ ft$/$cfm is operating at steady state with an inlet

flow of $10$ ft$3/$min $($or cfm$).$ At time t $=0,$ the flow is

suddenly increased to $100$ ft$3/$min for $0\mathrm{.}1$ min by adding

an additional $9$ ft$3$ of water to the tank uniformly over

a period of $0\mathrm{.}1$ min. Plot the response in tank level and

compare with the impulse response