1. The problem is to evaluate the magnitude of the complex number $\frac{5i}{3 - i}$.\n\n2. Recall that the magnitude of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$. Also, the magnitude of a quotient is the quotient of the magnitudes: $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$.\n\n3. First, find the magnitude of the numerator $5i$. Since $5i = 0 + 5i$, its magnitude is $|5i| = \sqrt{0^2 + 5^2} = 5$.\n\n4. Next, find the magnitude of the denominator $3 - i$. Here, $a=3$ and $b=-1$, so $|3 - i| = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}$.\n\n5. Now, compute the magnitude of the quotient:\n$$\left|\frac{5i}{3 - i}\right| = \frac{|5i|}{|3 - i|} = \frac{5}{\sqrt{10}}.$$\n\n6. To rationalize the denominator, multiply numerator and denominator by $\sqrt{10}$:\n$$\frac{5}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{5\sqrt{10}}{10} = \frac{\sqrt{10}}{2}.$$\n\n7. Therefore, the magnitude is $\boxed{\frac{\sqrt{10}}{2}}$.