A sign will be hung from the end of a thin pole, supported by a single cable. Your design firm has brainstormed six scenarios. In scenarios A, B, and D, the cable is attached midway between the pole's midpoint and end. In scenario C, the cable is attached at the pole's midpoint. In scenarios E and F, the cable is attached at the pole's end.

To provide a precise answer, it would be beneficial to have the diagrams or additional information about the scenarios mentioned (A, B, C, D, E, and F). However, given the descriptions provided, we can discuss some general principles and consider the mechanics involving tension, torque, and the center of gravity that will play a role in each scenario.

In general, the tension in the cable, the bending moment on the pole, and the stability of the sign will depend on where the cable is attached.

For scenarios A, B, and D - since the cable is attached between the midpoint and end of the pole, this should provide a decent amount of support to prevent the pole from bending due to the weight of the sign. However, exactly where between the midpoint and end can greatly affect the stability and the force on the cable. Closer to the midpoint means more support but a greater moment arm for the force exerted by the pole's weight, translating to more tension needed in the cable.

In scenario C - the cable is attached to the midpoint of the pole which will provide symmetry in terms of distributed forces, potentially resulting in a smaller bending moment in the pole as the load on the cable will be directly supporting the weight of the pole and sign. This is beneficial for the pole's integrity but may require a stronger cable since it has to fully counteract the weight of the sign without any assistance from the pole itself acting as a lever.

For scenarios E and F - the cable is attached to the end of the pole, which is very likely the least preferred scenario for stability. This setup can ensure that the cable takes most of the weight of the sign, but it will put a significant bending moment on the end of the pole, likely causing much more stress on the pole material and potentially leading to bending or failure.

In summary, the decision about where to locate the cable should account for the dimensions and material of the pole, the weight of the sign, the aesthetic needs, as well as safety considerations. A smart design would ideally minimize stress and maximize stability.