sove this please$($a$)$ A thermometer of first order characteristics having a time constant of $5$ seconds is at a steady state

temperature of $30C.$ At time $t=0,$ the thermometer is placed in a temperature bath maintained

at $90C.$ Determine the time needed for the thermometer to read $70C.$

$($b$)$ The delayed time of a first order flow system having process gain $K_p$ is $5$ units, and time constant

${}_p$ is $3$ seconds. Write down the transfer function, $G(s)$ of the delayed system if transportation lag

is $0\mathrm{.}5$ second.

$($c$)$ A thermometer having $1$st order dynamics with a time constant of $10$ seconds is placed in a

temperature bath maintained at $100C.$ After the thermometer reaches steady state with the bath.

it is then suddenly placed to another bath maintained at $120C$ and kept there for $1$ minute after

which it is immediately returned to the earlier bath. Deduce the expression of temperature profile

of the thermometer. Draw the dynamic behaviour of the thermometer.