Answer: C

Small Hint:

The wire consists of two common external tangent segments and one exposed arc on each circle

Big Hint:

Use radii $3$ and $9$; each tangent segment has length $\sqrt{12^2-6^2}$

Solution:

The centers are $12$ inches apart and their radii differ by $6.$ Thus each common external tangent segment has length $$\sqrt{12^2-6^2}=6\sqrt3,$$ contributing $12\sqrt3$ in all. The geometry of the tangent lines leaves a $120^\circ$ exposed arc on the radius-$3$ circle and a $240^\circ$ exposed arc on the radius-$9$ circle. Their lengths total $$\begin{aligned} &\frac{120}{360}(2\pi\cdot3)\\ &\quad{}+\frac{240}{360}(2\pi\cdot9)\\ &\qquad=2\pi+12\pi\\ &\qquad=14\pi. \end{aligned}$$ Hence the wire length is $12\sqrt3+14\pi.$

Thus, the correct answer is C.