Final answer:
To ensure the resultant force on the stake is vertical, the force P must be equal to the desired vertical force, since the vertical component of P on one rope is P × sin(30°). Consequently, the magnitude of the resultant force is simply P.
Step-by-step explanation:
To approach the trigonometry problem where a stake is pulled by two ropes with force P at an angle of 30°, we should consider the components of the forces in both the x (horizontal) and y (vertical) directions. Assuming the situation is symmetrical with two equal forces pulling at the same angle, the vertical components will add up to the resultant vertical force, and the horizontal components should cancel each other out for the resultant force to be strictly vertical.
For (a), to find the magnitude of P so that the resultant is vertical, let's denote the vertical component of one of the forces as Py. Since sin(30°) = 0.5, it means Py = P × 0.5. For two ropes, the total vertical component is 2Py = 2(P × 0.5) = P. Thus, the magnitude of one force P needed to keep the resultant vertical is the same as the total vertical force required.
For (b), the corresponding magnitude of the resultant force would be twice the force exerted by one rope in the vertical direction since there are two ropes. So, the resultant force R is R = 2 × Py = 2 × (P × sin(30°)) = 2 × (P × 0.5) = P.
To summarize, you'd set the magnitude of force P such that the vertical components of the forces from both ropes add up to the required vertical force, and by symmetry, the horizontal components cancel.