In the idealized system shown at the bottom, three forces are applied. Assume $=25.$

a$.[10$ points$]$ Using a graphical approach $($i$.$e$.,$ a diagram and geometry drawn by hand or using software to produce and measure the resultant, and not using component addition or trigonometry$)$ solve for the magnitude of $P$ that yields a vertical resultant force on the trolley, and find the magnitude of that resultant force. Ans. $P=277[$lb$],R=350[$lb$].$

b$.[10$ points$]$ Using trigonometry $($i$.$e$.,$ not using a graphical solution or component addition$),$ solve for the magnitude of $P$ that yields a vertical resultant force on the trolley. Also, solve for the magnitude of resultant force.

c$.[10$ points$]$ Using component addition $($i$.$e$.,$ not using a graphical approach or trigonometry$),$ solve for the magnitude of $P$ that yields a vertical resultant force on the trolley. Also, solve for the magnitude of resultant force.

d$.[5$ points$]$ What is significant about the case in which the resultant is vertical in terms of how the trolley will behave upon application of these forces? You do not need to use the SMART method for this part of the problem.