Planetary orbits. The formula T(x)=x^{\frac{3}{2}} calculates the time in

tabita57i 2021-09-22 Answered
Planetary orbits. The formula \(\displaystyle{T}{\left({x}\right)}={x}^{{{\frac{{{3}}}{{{2}}}}}}\) calculates the time in years that it takes a planet to orbit the sun if the planet is x times farther from the sun than earth is.
a) Find the inverse of T
b) What does the inverse of T calculate?
Given:
The time in years that it takes a planet to orbit the sun if the planet is x times farther from the sun than earth is given by \(\displaystyle{T}{\left({x}\right)}={x}^{{{\frac{{{3}}}{{{2}}}}}}\)

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Expert Answer

Tuthornt
Answered 2021-09-23 Author has 22847 answers
Formula used:
The formula \(\displaystyle{T}={f{{\left({x}\right)}}}={x}^{{{\frac{{{3}}}{{{2}}}}}}\) which converts planet orbital radius to orbital period. If \(\displaystyle{T}={f{{\left({x}\right)}}}\) and f(x) is a bijective mapping (both one-to-one and surjective) then its inverse exists and thus \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}\).
Calculation:
a) Consider the given formula \(\displaystyle{T}={f{{\left({x}\right)}}}={x}^{{{\frac{{{3}}}{{{2}}}}}}\) with \(\displaystyle{x}\geq{0}\), which is an increasing function satisfying bijectivity. Now, raising both the sides to power \(\displaystyle{\frac{{{2}}}{{{3}}}}\) we get \(\displaystyle{T}^{{{\frac{{{2}}}{{{3}}}}}}={\left({x}^{{{\frac{{{3}}}{{{2}}}}}}\right)}^{{{\frac{{{2}}}{{{3}}}}}}\). This gives, \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}={T}^{{{\frac{{{2}}}{{{3}}}}}}\)
b) From part (a) we have seen \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}={T}^{{{\frac{{{2}}}{{{3}}}}}}\). Thus inverse of T calculates orbit radius as a function of orbital period.
Conclusion:
Thus, we found out that \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}={T}^{{{\frac{{{2}}}{{{3}}}}}}\) is the inverse of T and it calculates orbit radius as a function of orbital period. We can also explain it in another way. If planet orbital period function is represented by \(\displaystyle{\left({x}\right)}={x}^{{{\frac{{{3}}}{{{2}}}}}}\), its inverse can be expressed as \(\displaystyle{T}^{{-{1}}}{\left({x}\right)}={x}^{{{\frac{{{2}}}{{{3}}}}}}\). For T(x), the argument x represents the number of times the planet with orbital period T is farther from sun than Earth is.
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