3. [5] Let $C$ be the circle of radius 12 centered at $(0,0)$. What is the length of the shortest path in the plane between $(8 \sqrt{3}, 0)$ and $(0,12 \sqrt{2})$ that does not pass through the interior of $C$ ?
Answer: $12+4 \sqrt{3}+\pi$ The shortest path consists of a tangent to the circle, a circular arc, and then another tangent. The first tangent, from $(8 \sqrt{3}, 0)$ to the circle, has length $4 \sqrt{3}$, because it is a leg of a 30-60-90 right triangle. The $15^{\circ}$ arc has length $\frac{15}{360}(24 \pi)$, or $\pi$, and the final tangent, to $(0,12 \sqrt{2})$, has length 12 .