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

tabita57i

tabita57i

Answered question

2021-09-22

Planetary orbits. The formula T(x)=x32 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 T(x)=x32

Answer & Explanation

Tuthornt

Tuthornt

Skilled2021-09-23Added 107 answers

Formula used:
The formula T=f(x)=x32 which converts planet orbital radius to orbital period. If T=f(x) and f(x) is a bijective mapping (both one-to-one and surjective) then its inverse exists and thus x=f1(T).
Calculation:
a) Consider the given formula T=f(x)=x32 with x0, which is an increasing function satisfying bijectivity. Now, raising both the sides to power 23 we get T23=(x32)23. This gives, x=f1(T)=T23
b) From part (a) we have seen x=f1(T)=T23. Thus inverse of T calculates orbit radius as a function of orbital period.
Conclusion:
Thus, we found out that x=f1(T)=T23 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 (x)=x32, its inverse can be expressed as T1(x)=x23. 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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