Problem

The diagram shows the curve with equation $y=\sqrt{6x+5}$. The shaded region is bounded by the curve, the $x$-axis and the lines $x=a$ and $x=2a$, where $a$ is a positive constant.

The shaded region is rotated through $360^\circ$ about the $x$-axis to form a solid. The solid has volume, $V$, such that $V\ge 46\pi$.

(a) Show that $9a^2+5a-46\ge 0$.

(b) Find the range of possible values of $a$.