The volume of the solid of revolution formed by rotating the curve \(y = \frac{a}{x}\) from \(x = 1\) to \(x = 3\) about the x-axis is given by:

\(V = \pi \int_{1}^{3} \left( \frac{a}{x} \right)^2 \, dx\)

\(= \pi \int_{1}^{3} \frac{a^2}{x^2} \, dx\)

Integrating \(\frac{a^2}{x^2}\) gives:

\(= \pi \left[ -\frac{a^2}{x} \right]_{1}^{3}\)

\(= \pi \left( -\frac{a^2}{3} + \frac{a^2}{1} \right)\)

\(= \pi \left( a^2 - \frac{a^2}{3} \right)\)

\(= \pi \left( \frac{2a^2}{3} \right)\)

We are given that this volume is \(24\pi\), so:

\(\pi \left( \frac{2a^2}{3} \right) = 24\pi\)

\(\frac{2a^2}{3} = 24\)

\(2a^2 = 72\)

\(a^2 = 36\)

\(a = 6\)