A violin manufacturer is designing a new violin, and a Lituus curve is being used to model the neck.

The equations of the curve are

for $\frac{\pi }{60}\le t\le \frac{\pi }{6}$.

The manufacturer requires to know the $x$ coordinate of the point $B$, which is the intersection between the $x$ axis and the tangent to the curve at the point $A$, where $A$ has $x$-coordinate equal to $0$.

Find the exact $x$ coordinate of the point $B$.

The answer is of the form $\sqrt{n\pi }$ for some $n\in \mathbb{N}$. Give the value of $n$.