Question 1023909
Stated in formulas, rather than words,
Kepler's Third Law states that
if {{{T}}}= time required for a planet to complete one orbit around the sun (the period, that is, the length of one planetary year), and
{{{d}}}= average distance of the planet from the sun,
then {{{T^2=k*d^3}}} ,
where {{{k}}} is the proportionality constant.
So, for Earth, {{{T[Earth]^2=k*d[Earth]^3}}}
For the planet Earth, assume {{{d[Earth]}}} = {{{9*10^6}}} {{{miles}}} an {{{T[Earth]= 365}}}{{{days}}} .
Since we are going to measure times in Earth days and distances in miles
(or in whatever units we want, but using the same units all along),
we do not need to keep writing the units with the calculations.
NOTE: I did not solve for {{{k}}} and I did not use the given average distance between Earth and the sun, because it would only complicate the calculations.
If your teacher insist that you do it, I am sorry for both of you.
 
a) They tell us that {{{d[Mars]}}}={{{approximately}}}{{{1.5*d[Earth]}}} ,
and according to Kepler's Third Law
{{{T[Mars]^2=k*d[Mars]^3}}} , so
{{{T[Mars]^2=k*(1.5*d[Earth])^3}}}
{{{T[Mars]^2=k*1.5^3*d[Earth]^3}}}
{{{T[Mars]^2=T[Earth]^2*1.5^3}}}
{{{T[Mars]^2=(k*d[Earth]^3)*1.5^3}}}
{{{T[Mars]^2=365^2*1.5^3}}}
{{{T[Mars]^2=133225*3.375}}}
{{{T[Mars]^2=449634.375}}}
{{{T[Mars]=sqrt(449634.375)}}}
{{{T[Mars]=about671}}} (rounded).
So, the period of Mars is approximately 671 days.
 
b) They tell us that {{{T[Venus]=223}}} ,
and according to Kepler's Third Law
{{{T[Venus]^2=k*d[Venus]^3}}} , and {{{T[Venus]^2=k*d[Venus]^3}}} ,
so with distances in miles this time
{{{223^2=k*d[Venus]^3}}} , and {{{365^2=k*(93*10^6)^3}}} .
Dividing one equation by the other,
{{{223^2/365^2=k*d[Venus]^3/(k*(93*10^6)^3)}}}
{{{49729/133225=d[Venus]^3/(93^3*10^18)}}}
{{{49729/133225=d[Venus]^3/(804357*10^18)}}}
{{{49729*804357*10^18/133225=d[Venus]^3}}}
{{{49729*804357*10^18/133225=d[Venus]^3}}}
{{{300243*10^18=d[Venus]^3}}}
{{{root(3,300243*10^18)=d[Venus]}}}
{{{d[Venus]=root(3,300243)*root(3,10^18)}}}
{{{d[Venus]=67*10^6}}} (rounded).
So, the the average distance of Venus from the sun, is approximately 67 x 10^6 miles.