1. **State the problem:** We have a right triangle with angles 30\degree, 60\degree, and 90\degree. The side adjacent to the 30\degree angle is $3 - \sqrt{6}$ miles, and the side opposite the 30\degree angle is $c$. We need to find $c$ in simplest radical form. 2. **Recall the properties of a 30\degree-60\degree-90\degree triangle:** The sides are in the ratio $1 : \sqrt{3} : 2$, where: - The side opposite 30\degree is the shortest side, call it $x$. - The side opposite 60\degree is $x\sqrt{3}$. - The hypotenuse is $2x$. 3. **Identify the given side:** The side adjacent to the 30\degree angle is opposite the 60\degree angle, so it corresponds to $x\sqrt{3}$. Given that side is $3 - \sqrt{6}$, so: $$x\sqrt{3} = 3 - \sqrt{6}$$ 4. **Solve for $x$:** $$x = \frac{3 - \sqrt{6}}{\sqrt{3}}$$ 5. **Simplify $x$ by rationalizing the denominator:** $$x = \frac{3 - \sqrt{6}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{(3 - \sqrt{6})\sqrt{3}}{3}$$ 6. **Distribute $\sqrt{3}$ in the numerator:** $$x = \frac{3\sqrt{3} - \sqrt{6}\sqrt{3}}{3} = \frac{3\sqrt{3} - \sqrt{18}}{3}$$ 7. **Simplify $\sqrt{18}$:** $$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ 8. **Substitute back:** $$x = \frac{3\sqrt{3} - 3\sqrt{2}}{3}$$ 9. **Cancel the common factor 3:** $$x = \frac{\cancel{3}\sqrt{3} - \cancel{3}\sqrt{2}}{\cancel{3}} = \sqrt{3} - \sqrt{2}$$ 10. **Recall $c$ is the side opposite 30\degree, which equals $x$:** $$c = x = \sqrt{3} - \sqrt{2}$$ **Final answer:** $$\boxed{\sqrt{3} - \sqrt{2}}$$ miles