A thermometer requires $1$ min to indicate $98\%$ of the response to a step input. If the

thermometer is placed in a bath with a water temperature $10\backslash$deg $,$ the ultimate tempera$-$

ture shown in the thermometer is $7\mathrm{.}5\backslash$deg $.$ Assume that the thermometer is a first$-$order

system

H$($s$)=($K$)/(\backslash$tau s$+1)$

$($a$)$ Find the time constant $\backslash$tau $.$

$($b$)$ Find the value of K$.$