Final answer:
The plate supported by a ball-and-socket joint at A, a roller joint at B, and a cable at C has a total of five unknown support reactions: three forces from joint A, one force from joint B, and one force from the cable at C.
Step-by-step explanation:
The number of unknown support reactions at a plate can be determined by considering the type of supports and the reactions they provide. A ball-and-socket joint at A allows rotation in all directions but prevents translation, therefore, it has three unknown force reactions but no moment. A roller joint at B allows movement in one direction but resists motion in the perpendicular plane, leading to only one unknown force reaction and no moment. Lastly, a cable at C can only exert a force along its length, adding an additional unknown force reaction. Thus, the total number of unknown support reactions consists of three forces from the ball-and-socket joint (A), one force from the roller joint (B), and one force from the cable (C), making a total of five forces.