Answer
$b=-2.4$
Work Step by Step
Firstly differentiate $PV^b=C$ with respect to time $t$ using product rule and chain rule.
We get, $\dfrac{dP}{dt}V^b+bV^{b-1}P\dfrac{dV}{dt}=0$
$\implies\dfrac{dP}{dt}V+bP\dfrac{dV}{dt}=0$
Now substitute $P = 25 kPa$, $\dfrac{dP}{dt} = 12 \dfrac{kPa}{min}$, $V = 100 cm^3$, and
$\dfrac{dV}{dt}= 20 \dfrac{cm^3}{min}$ in $\dfrac{dP}{dt}V+bP\dfrac{dV}{dt}=0$.
$1200+b\cdot25\cdot20=0$
$\implies 500b=-1200$
$\implies b=\dfrac{-1200}{500}=-2.4$
