Answer
$t = 16.4~s$
Work Step by Step
We can express the angular speed in units of $rad/s$:
$\omega_C = (100~rev/min)(\frac{2\pi~rad}{1~rev})(\frac{1~min}{60~s}) = 10.47~rad/s$
We can find the linear speed on the rim of wheel C:
$v = \omega_C~r_C$
$v = (10.47~rad/s)(0.25~m)$
$v = 2.618~m/s$
We can find the angular speed of wheel A:
$\omega_A = \frac{v}{r_A}$
$\omega_A = \frac{2.618~m/s}{0.10~m}$
$\omega_A = 26.18~rad/s$
We can find the required time to reach this angular speed:
$\omega_A = \omega_0+\alpha~t$
$\omega_A = 0+\alpha~t$
$t = \frac{\omega_A}{\alpha}$
$t = \frac{26.18~rad/s}{1.6~rad/s^2}$
$t = 16.4~s$
