Discussion
If the A.M. of \(a\) and \(c\) is 16 and if \(a = 8\) , then the G.M. of \(a\) and \(c\) is
Answer: A
The G.M. is \(8\sqrt{3}\).
The arithmetic mean gives the second number, then the geometric mean is the square root of the product:
\[ \begin{aligned} \frac{a + c}{2} &= 16, \quad c = 32 - 8 = 24 \\ \text{G.M.} &= \sqrt{ac} = \sqrt{8 \times 24} \\ &= \sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3} \end{aligned} \]
A check: \(8\sqrt{3} \approx 13.9\), which is less than the A.M. 16, as the A.M.-G.M. inequality requires. \(4\sqrt{3}\) is half the answer, from using \(\sqrt{ac}/2\) or taking \(c = 6\).
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