#### Answer

(a) $\Delta \theta = (\frac{\omega_0~T}{2})~rad$
(b) The turbine rotates through 19000 revolutions as it coasts to a stop.

#### Work Step by Step

(a) The turbine coasts to a stop from $\omega_0~rad/s$. During this time, the average angular speed is $\frac{\omega_0~rad/s+0}{2}$ which is $\frac{\omega_0~rad/s}{2}$. We can find the angle through which the turbine rotates as it stops:
$\Delta \theta = (\frac{\omega_0~rad/s}{2})(T~s)$
$\Delta \theta = (\frac{\omega_0~T}{2})~rad$
(b) The turbine coasts to a stop from 3800 rpm. During this time, the average angular speed is $\frac{3800~rpm+0}{2}$ which is 1900 rpm. We can find the number of revolutions the turbine rotates through as it stops:
$\theta = (1900~rpm)(10~min) = 19000~rev$
The turbine rotates through 19000 revolutions as it coasts to a stop.