Answer
The rotational speed of the lathe chuck is $9.05~rad/s$
Work Step by Step
We can find the time it takes the cutting tool to move one inch:
$t = \frac{d}{v} = \frac{1~in}{0.080~in/s} = 12.5~s$
The lathe chuck must rotate 18 times in this time. We can find the angular speed:
$\omega = \frac{18\times 2\pi~rad}{12.5~s} = 9.05~rad/s$
The rotational speed of the lathe chuck is $9.05~rad/s$
