1. **State the problem:** We have triangle UVW with side $w=35$ inches opposite angle $W=154^\circ$, and angle $U=10^\circ$. We need to find the length of side $u$ opposite angle $U$. 2. **Find the missing angle:** The sum of angles in a triangle is $180^\circ$. $$m\angle V = 180^\circ - m\angle W - m\angle U = 180^\circ - 154^\circ - 10^\circ = 16^\circ$$ 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{u}{\sin U} = \frac{w}{\sin W}$$ 4. **Plug in known values:** $$\frac{u}{\sin 10^\circ} = \frac{35}{\sin 154^\circ}$$ 5. **Calculate sine values:** $$\sin 10^\circ \approx 0.1736$$ $$\sin 154^\circ = \sin (180^\circ - 154^\circ) = \sin 26^\circ \approx 0.4384$$ 6. **Solve for $u$:** $$u = \frac{35 \times \sin 10^\circ}{\sin 154^\circ} = \frac{35 \times 0.1736}{0.4384}$$ 7. **Simplify:** $$u = \frac{35 \times 0.1736}{0.4384} \approx \frac{6.076}{0.4384} \approx 13.86$$ 8. **Round to nearest inch:** $$u \approx 14$$ inches. **Final answer:** The length of side $u$ is approximately 14 inches.