Respuesta :
Answer:
Therefore the asteroids's orbital period is 7.45 years.
Explanation:
Kapler's third law:
The orbital period of a planet squared is directly proportional to the average distance of the planet from the sun cubed.
[tex]T^2\propto r^3[/tex]
T = orbital period of the planet
r = orbital radius of the planet
[tex]\therefore \frac{T_1^2}{T_2^2}=\frac{R_1^3}{R_2^3}[/tex]
Here [tex]T_1[/tex] = Orbital period of the asteroid
[tex]R_1[/tex]= Orbital radius of the asteroid
[tex]T_2=[/tex] Orbital period of Earth = 1 year
[tex]R_2[/tex] =Orbital radius of Earth
Given that the orbital radius of asteroid is 3.83 times of orbital radius of Earth.
[tex]R_1 = 3.83R_2[/tex]
[tex]\frac{R_1}{R_2}=3.83[/tex]
Therefore
[tex]\therefore \frac{T_1^2}{T_2^2}=\frac{R_1^3}{R_2^3}[/tex]
[tex]\Rightarrow ( \frac{T_1}{T_2})^2=(\frac{R_1}{R_2})^3[/tex]
[tex]\Rightarrow ( \frac{T_1}{T_2})^2=(3.83)^3[/tex]
[tex]\Rightarrow \frac{T_1}{T_2}=(3.83)^\frac32[/tex]
[tex]\Rightarrow {T_1=(3.83)^\frac32\times T_2[/tex]
[tex]\Rightarrow T_1= 7.45\times (1 \ year)[/tex] [[tex]T_2=[/tex] 1 year]
[tex]\Rightarrow T_1= 7.45 \ years[/tex]
Therefore the asteroids's orbital period is 7.45 years.
