Answer
The tangential velocity is 55.64 mi/s
Work Step by Step
We can express the angle $\theta$ in radians:
$\theta = (10.34'')\times \frac{1^{\circ}}{3600''}\times \frac{\pi~rad}{180^{\circ}} = 0.0000501297~rad$
We can find the tangential velocity in units of mi/year:
$v_t = \theta~r$
$v_t = (0.0000501297~rad)(35\times 10^{12}~mi/yr)$
$v_t = 1.7545407\times 10^9~mi/yr$
We can convert $v_t$ to units of mi/s:
$v_t = (1.7545407\times 10^9~mi/yr)\times \frac{1~yr}{(365)~ (24)~ (3600)~s}$
$v_t = 55.64~mi/s$
The tangential velocity is 55.64 mi/s
