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The circumference of the circle is $40.715$ inches. Sector PQ is $72^\circ$. That means that PQR is $\dfrac{137}{360}$ of the circumference. $40.715 \times \dfrac{137}{360}\approx15.4943$ inches.
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In a circle, we know that all the angles must have sum upto $360^\circ$. So for the angle we don't know we can write:
$$73^\circ+150^\circ+65^\circ+x^\circ=360^\circ\space\Longleftrightarrow\space x^\circ=360^\circ-\left(73^\circ+150^\circ+65^\circ\right)=72^\circ\tag1$$
Now, for the circumference of a circle we have:
$$\mathcal{S}=2\pi\cdot\text{r}\tag2$$
And, when we have a sector of the circle:
$$\mathcal{S}_\Delta=2\pi\cdot\frac{\theta^\circ}{360^\circ}\cdot\text{r}\tag3$$
So, in your example we have that:
$$\theta^\circ=x^\circ+65^\circ=72^\circ+65^\circ=137^\circ\tag4$$
So, the arc length will be:
$$\mathcal{S}_\text{PQR}=2\pi\cdot\frac{137^\circ}{360^\circ}\cdot6.48=\frac{1233\pi}{250}\approx15.4943349\space\text{inch}\tag5$$
Arc length = Radius * Angle subtended by the arc at the centre(in radian)