Question

select the correct answer.
the arithmetic mean (a) of two numbers (a and b) is given by the formula $a = \frac{a + b}{2}$, and their geometric mean (g) is given by $g = \sqrt{ab}$. their harmonic mean (h) is given by the formula $g = \sqrt{ah}$. which formula correctly gives h in terms of a and b?
\\(\bigcirc\\) a. $h = \frac{2ab}{a + b}$
\\(\bigcirc\\) b. $h = \sqrt{\frac{2ab}{a + b}}$
\\(\bigcirc\\) c. $h = \frac{a + b}{2ab}$
\\(\bigcirc\\) d. $h = \frac{ab}{2(a + b)}$

Explanation:

Step1: Substitute \(A\) and \(G\) into \(G = \sqrt{AH}\)

Given \(A=\frac{a + b}{2}\) and \(G=\sqrt{ab}\), substitute into \(G = \sqrt{AH}\): \(\sqrt{ab}=\sqrt{\frac{a + b}{2}\times H}\)

Step2: Square both sides of the equation

Squaring both sides gives \(ab=\frac{a + b}{2}\times H\)

Step3: Solve for \(H\)

Multiply both sides by \(2\) to get \(2ab=(a + b)H\), then divide both sides by \((a + b)\): \(H=\frac{2ab}{a + b}\)

Answer:

A. \(H=\frac{2ab}{a + b}\)