The fourth proportional to the first number, the second number, and 20 is 36. If the sum of the first two numbers is 14, find the geometric mean of the first two numbers.
\(3\sqrt{5}\)
Let the first number be \(a\) and the second number be \(b\).
The fourth proportional to \(a, b, 20\) is 36 means \(a : b = 20 : 36\), i.e. \(\frac{a}{b} = \frac{20}{36} = \frac{5}{9}\).
So let \(a = 5k\) and \(b = 9k\) for some constant \(k\).
Given \(a + b = 14\), so \(5k + 9k = 14 \Rightarrow 14k = 14 \Rightarrow k = 1\).
Hence \(a = 5\) and \(b = 9\).
Geometric mean \(= \sqrt{ab} = \sqrt{5 \times 9} = \sqrt{45} = 3\sqrt{5}\).
Hence, the geometric mean of the first two numbers is \(3\sqrt{5}\).
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