1. **State the problem:** We need to solve for side $c$ in a triangle using the Law of Cosines. 2. **Law of Cosines formula:** $$c^2 = a^2 + b^2 - 2ab\cos C$$ where $a$ and $b$ are sides of the triangle, and $C$ is the angle opposite side $c$. 3. **Given values:** - $a = 3$ - $b = 10$ - $C = 34^\circ$ 4. **Substitute the known values into the formula:** $$c^2 = 3^2 + 10^2 - 2 \times 3 \times 10 \times \cos 34^\circ$$ 5. **Calculate each term:** - $3^2 = 9$ - $10^2 = 100$ - $2 \times 3 \times 10 = 60$ - $\cos 34^\circ \approx 0.8290$ 6. **Evaluate the expression:** $$c^2 = 9 + 100 - 60 \times 0.8290 = 109 - 49.74 = 59.26$$ 7. **Find $c$ by taking the square root:** $$c = \sqrt{59.26} \approx 7.7$$ **Final answer:** $$c \approx 7.7$$