I assumed that in order to minimize the force that the child would have to apply, the child vector would have a magnitude of zero and the you vector would have all of the force to lift the bag, but then what would be the value of beta?

You wish to help the child carry the backpack. Find the magnitude of force you should apply to the backpack with weight 31.1 lb and the angle to minimize the amount of force the child needs to provide if a is fixed at 35.2.