1. **State the problem:** We are given a triangle UVW with angles $\angle U = 30^\circ$, $\angle V = 104^\circ$, and $\angle W = 46^\circ$. Side $UV = 11$ units is opposite $\angle W$. We need to find sides $u$ (opposite $\angle U$) and $v$ (opposite $\angle V$). 2. **Use the Law of Sines:** The Law of Sines states that $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 3. **Assign known values:** Here, side $UV = 11$ is opposite $\angle W = 46^\circ$, so $$c = 11, \quad C = 46^\circ.$$ We want to find $u$ opposite $\angle U = 30^\circ$ and $v$ opposite $\angle V = 104^\circ$. 4. **Set up ratios:** Using Law of Sines, $$\frac{u}{\sin 30^\circ} = \frac{v}{\sin 104^\circ} = \frac{11}{\sin 46^\circ}.$$ 5. **Calculate $u$:** $$u = \frac{11}{\sin 46^\circ} \times \sin 30^\circ.$$ Calculate the sines: $\sin 46^\circ \approx 0.7193$, $\sin 30^\circ = 0.5$. So, $$u = \frac{11}{0.7193} \times 0.5 = 15.29 \times 0.5 = 7.645.$$ Rounded to nearest tenth, $$u \approx 7.6.$$ 6. **Calculate $v$:** $$v = \frac{11}{\sin 46^\circ} \times \sin 104^\circ.$$ Calculate $\sin 104^\circ \approx 0.9703$. So, $$v = \frac{11}{0.7193} \times 0.9703 = 15.29 \times 0.9703 = 14.83.$$ Rounded to nearest tenth, $$v \approx 14.8.$$ **Final answers:** $$u = 7.6, \quad v = 14.8, \quad m\angle W = 46^\circ.$$