1. **State the problem:** We need to find the length of the cable (side $AB$) in triangle $ABC$ where $AC=105$ m, angle $\angle BAC=12^\circ$, and the hill's inclination angle $\angle ACB=58^\circ$. 2. **Identify the angles:** The triangle's angles are $\angle BAC=12^\circ$, $\angle ACB=58^\circ$, and the remaining angle $\angle ABC$ can be found by the triangle angle sum rule: $$\angle ABC = 180^\circ - 12^\circ - 58^\circ = 110^\circ$$ 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{AB}{\sin(58^\circ)} = \frac{AC}{\sin(110^\circ)} = \frac{BC}{\sin(12^\circ)}$$ We want to find $AB$, so: $$AB = AC \times \frac{\sin(58^\circ)}{\sin(110^\circ)}$$ 4. **Calculate the sine values:** $$\sin(58^\circ) \approx 0.8480$$ $$\sin(110^\circ) \approx 0.9397$$ 5. **Calculate $AB$:** $$AB = 105 \times \frac{0.8480}{0.9397}$$ 6. **Simplify the fraction:** $$\frac{0.8480}{0.9397} \approx 0.9029$$ 7. **Final calculation:** $$AB \approx 105 \times 0.9029 = 94.8$$ **Answer:** The length of the cable required is approximately **94.8 m**.