Answer
(a) 5.0 years
(b) 2.2 years
(c) 9.5 years
Work Step by Step
(a) We can find the time $t'$ according to Mission Control:
$t' = \frac{d}{v}$
$t' = \frac{4.5~ly}{0.90~c}$
$t' = \frac{4.5~y}{0.90}$
$t' = 5.0~years$
(b) We can find the time $t_0$ that passes in the astronaut's reference frame:
$t' = \gamma~t_0$
$t_0 = \frac{t'}{\gamma}$
$t_0 = t'~\sqrt{1-\frac{v^2}{c^2}}$
$t_0 = t'~\sqrt{1-\frac{(0.90~c)^2}{c^2}}$
$t_0 = (5.0~y)~\sqrt{1-(0.90)^2}$
$t_0 = 2.2~years$
(c) The time required for the radio message to travel a distance of 4.5 light years is 4.5 years. The total time that passes is $5.0~y+4.5~y$ which is $9.5~years$
