Answer:
0.05806
Explanation:
[tex]m_x[/tex] = Mass of asteroid x
[tex]m_y[/tex] = Mass of asteroid y
[tex]r_x[/tex] = Distance from asteroid x = 140 km
[tex]r_y[/tex] = Distance from asteroid y = 581 km
m = Mass of asteroid
Force of gravity between asteroid x and the astronaut
[tex]F_1=\frac{Gm_xm}{r_x^2}\\\Rightarrow F_1=\frac{Gm_xm}{140^2}[/tex]
Force of gravity between asteroid x and the astronaut
[tex]F_2=\frac{Gm_ym}{r_y^2}\\\Rightarrow F_2=\frac{Gm_ym}{581^2}[/tex]
Here these two forces are equal as they are in equilibrium
[tex]\frac{Gm_xm}{140^2}=\frac{Gm_ym}{581^2}\\\Rightarrow \frac{m_x}{140^2}=\frac{m_y}{581^2}\\\Rightarrow \frac{m_x}{m_y}=\frac{140^2}{581^2}\\\Rightarrow \frac{m_x}{m_y}=0.05806[/tex]
The ratio of the masses of the asteroid is 0.05806
