Answer
(a) The passenger will be 30 years old when the spaceship reaches the destination.
(b) The signal reaches the Earth in the year 3420.
Work Step by Step
(a) We can find the time to reach the destination, according to a clock on the Earth:
$\Delta t = \frac{710~ly}{0.9999~c} \approx 710 ~years$
Let $\Delta t_0$ be the time measured on the spaceship.
$\Delta t = \frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}}$
$\Delta t_0 = \Delta t~\sqrt{1-\frac{v^2}{c^2}}$
$\Delta t_0 = (710~years)~\sqrt{1-\frac{(0.9999~c)^2}{c^2}}$
$\Delta t_0 = 10.0~years$
The passenger will be 30 years old when the spaceship reaches the destination.
(b) According to a clock on the Earth, the spaceship takes about 710 years to arrive. Then the signal will take 710 years to travel 710 light years. The total time that passes is 1420 years. Therefore, the signal reaches the Earth in the year 3420.
