Answer
The astronaut's mass is 120 kg
Work Step by Step
We can find the force constant $k$ of the spring.
$T = 2\pi~\sqrt{\frac{m}{k}}$
$k = \frac{(2\pi)^2~m}{T^2}$
$k = \frac{(2\pi)^2~(42.5~kg)}{(1.30~s)^2}$
$k = 992.8~N/m$
We can use the period when the astronaut is sitting in the chair to find the total mass of the chair and the astronaut.
$T = 2\pi~\sqrt{\frac{m}{k}}$
$m = \frac{T^2~k}{(2\pi)^2}$
$m = \frac{(2.54~s)^2(992.8~N/m)}{(2\pi)^2}$
$m = 162.2~kg$
To find the astronaut's mass $m_a$, we need to subtract the mass of the chair.
$m_a = 162.2~kg-42.5~kg = 120~kg$
The astronaut's mass is 120 kg