An oven used to "bake" semiconductor chips in their manufacturing process exhibits first$-$order

dynamics without time delay, relating the oven temperature, $T(t),$ to changes in the oven heater

power input, $P(t),$ as shown below:

$\frac{T^{\text{'}}(s)}{P^{\text{'}}(s)}=\frac{K}{s+1}$

The process is at steady$-$state when an engineer changes the power input stepwise from $1kW$ to

$2kW.$ The engineer notes the following:

The oven temperature initially is $100C$ and steady.

Four minutes after changing the power input, the oven temperature is $400C.$

Thirty minutes later, the oven temperature is $500C$ and steady.

a$.$ What is the gain of this process, with units?

b$.$ What is the time constant of this process, with units?