Question 970839
this is basically a question in direct variation.


the problem is not difficult, but the arithmetic is messy.


the formula for direct variation is y = k*x.


if p = orbital period and m = mean orbital radius, then the equation of y = k*x becomes:


p^2 = k * m^3


this means that the orbital period squared is directly proportional to the mean orbital radius cubed.


with eath, we'll set 1 year as 12 months to keep the values consistent with each other.


the equation for earth becomes:


(12)^2 = k * (150 * 10^6)^3


we solve for k to get:


k = (12)^2 / (150 * 10^6)^3


now we use that value of k to solve for the mean orbital radius of venus.


the formula of p^2 = k * m^3 becomes:


(7.38)^2 = (12)^2 / (150 * 10^6)^3 * m^3


solve for m^3 to get:


m^3 = (7.38)^2 / (12)^2 * (150 * 10^6)^3


this is the messy part.


it's best to just use your calculator to solve it.


you will get:


m^3 = 1.276509376 * 10^24


take the cube root of both sides of this equation to get:


m = 108477916.7 km.


this is equivalent to 108.4779167 * 10^6 km.


this solution is consistent with what i researched for the mean orbital radius of venus and of earth.


earth is 150 * 10^6 and venus is approximately 108 * 10^6.


this confirms the solution is correct.


here's what i found for mean orbital radius of the earth.


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The radius of the earth's orbit around the sun (assumed to becircular) is 1.50 10^8 km, and the earth travels around this orbit in 365 days
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1.50 * 10^8 km is the same at 150 * 10^6 km.


here's what i found for mean orbital radius of venus.


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Venus orbits the Sun at an average distance of about 108,000,000 km and completes an orbit every 224.65 days. 
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108,000,000 is equivalent to 108 * 10^6.
224.65 days is equivalent to 7.38 months.