 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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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.