Since we are given by G=sqrt(AH):
Rewriting;
G=sqrt(AH)
//square both sides//
G^2=[sqrt(AH)]^2
G^2=AH
A=G^2/H
G=sqrt(AH)
//square both sides//
G^2=[sqrt(AH)]^2
G^2=AH
H=G^2/A
Thus, the answer is the first option which has
A=G^2/H and
H=G^2/A
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMS-AM-GM-HM), is an inequality of the root-mean square, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers that says:
with equality if and only if . This inequality can be expanded to the power mean inequality.
As a consequence we can have the following inequality: If are positive reals, then with equality if and only if ; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving.