Question

if the geometric mean of a and 65 is 15√78,
find the value of a.
a =

Explanation:

Step1: Recall the geometric mean formula

The geometric mean of two numbers \(x\) and \(y\) is \(\sqrt{xy}\). Here, \(x = a\), \(y = 65\), and the geometric mean is \(15\sqrt{78}\). So, \(\sqrt{a\times65}=15\sqrt{78}\).

Step2: Square both sides of the equation

Squaring both sides gives \(a\times65=(15\sqrt{78})^2\).
Since \((15\sqrt{78})^2 = 15^2\times78=225\times78\), then \(a\times65 = 225\times78\).

Step3: Solve for \(a\)

\(a=\frac{225\times78}{65}\).
Simplify \(\frac{78}{65}=\frac{6}{5}\), so \(a = 225\times\frac{6}{5}\).
\(225\div5 = 45\), and \(45\times6=270\).

Answer:

\(270\)