Question 1208014: Kepler’s Third Law of Planetary Motion states that the square of the period of revolution T of a planet varies directly with the cube of its mean distance a from the Sun.If the mean distance of Earth from the Sun is 93 million miles, what is the mean distance of the planet Mercury from the Sun, given that Mercury has a “year”of 88 days?
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39618)   (Show Source):
You can put this solution on YOUR website! -------------------------------------------------------
that the square of the period of revolution T of a planet varies directly with the cube of its mean distance a from the Sun.
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k, the variation constant,
d, the distance,
the language in the description means 
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Answer: 36 million miles (approximate)
Explanation
T = number of days
d = average distance, in millions of miles, the certain planet is from the sun
We can paraphrase the info in the instructions to say:
"Square of T varies directly with the cube of d"
and gives this equation
T^2 = k*d^3
for some constant of variation k.
Why c isn't used instead of k, I'm not sure, but this is common convention used in many textbooks.
On Earth there are roughly 365 days in a year.
Of course the reality is a bit more complicated when considering leap years, but I'll try to keep things relatively simple.
T = 365 pairs up with d = 93
Use these values to find k.
T^2 = k*d^3
365^2 = k*93^3
k = (365^2)/(93^3)
k = 0.165629192013 approximately
We'll use that k value, along with T = 88 to find its paired d value for the planet Mercury.
T^2 = k*d^3
88^2 = 0.165629192013*d^3
7744 = 0.165629192013*d^3
d = cubeRoot( 7744/0.165629192013 )
d = ( 7744/0.165629192013 )^(1/3)
d = 36.025456026483 approximately
d = 36
The average distance from the Sun to Mercury is roughly 36 million miles.
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This is entirely optional, but with science based questions, I find it good practice to verify with reputable sources.
https://science.nasa.gov/mercury/facts/
Quote from the page: "From an average distance of 36 million miles (58 million kilometers), Mercury is 0.4 astronomical units away from the Sun."
Notes:
AU = astronomical units
1 AU = 93 million miles approximately
0.4 AU = 0.4*(93 million miles) = 37.2 million miles, which is fairly close to the 36 mentioned.
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