To determine the correct relationship between the period of revolution ([tex]$T$[/tex]) and the mean distance from the Sun ([tex]$R$[/tex]), we refer to Kepler's Third Law of Planetary Motion. This law states that the square of a planet's orbital period ([tex]$T^2$[/tex]) is proportional to the cube of the semi-major axis of its orbit ([tex]$R^3$[/tex]).
Kepler's Third Law is mathematically expressed as:
[tex]\[ T^2 \propto R^3 \][/tex]
This means that if you know the mean distance from the Sun ([tex]$R$[/tex]), you can determine that:
[tex]\[ T^2 = k \cdot R^3 \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
Given the options:
A. [tex]\( R \)[/tex]
B. [tex]\( R^2 \)[/tex]
C. [tex]\( R^3 \)[/tex]
D. [tex]\( R^4 \)[/tex]
E. 5
The correct answer, according to Kepler's Third Law, is:
[tex]\[ \boxed{R^3} \][/tex]
Thus, [tex]\( T^2 \)[/tex] varies directly as [tex]\( R^3 \)[/tex]. Therefore, the correct choice is C. [tex]\( R^3 \)[/tex].
