1. **State the problem:** We need to find the length of the cable, which forms the hypotenuse of a right triangle with legs 12 feet (flagpole height) and 6 feet (distance from base). 2. **Formula used:** Use the Pythagorean theorem for right triangles: $$c = \sqrt{a^2 + b^2}$$ where $c$ is the hypotenuse (cable length), $a$ and $b$ are the legs. 3. **Apply values:** $$c = \sqrt{12^2 + 6^2} = \sqrt{144 + 36} = \sqrt{180}$$ 4. **Simplify the square root:** $$\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}$$ 5. **Estimate $\sqrt{5}$:** Since $2^2 = 4$ and $3^2 = 9$, $\sqrt{5}$ is between 2 and 3. More precisely, $\sqrt{5} \approx 2.2$ (since $2.2^2 = 4.84$ close to 5). 6. **Calculate approximate length:** $$c \approx 6 \times 2.2 = 13.2$$ 7. **Choose the correct answer:** Between the options 12.7 and 13.4, 13.2 is closer to 13.4. **Final answer:** The length of the cable is approximately **13.4 feet**.