Serway & Jewett — Vectors: Problem 46

A pirate has buried his treasure on an island with five trees located at the points \((30.0\text{ m}, -20.0\text{ m})\), \((60.0\text{ m}, 80.0\text{ m})\), \((-10.0\text{ m}, -10.0\text{ m})\), \((40.0\text{ m}, -30.0\text{ m})\), and \((-70.0\text{ m}, 60.0\text{ m})\), all measured relative to some origin, as shown in Figure P3.46. His ship’s log instructs you to start at tree \(A\) and move toward tree \(B\), but to cover only one-half the distance between \(A\) and \(B\). Then move toward tree \(C\), covering one-third the distance between your current location and \(C\). Next move toward tree \(D\), covering one-fourth the distance between where you are and \(D\). Finally move toward tree \(E\), covering one-fifth the distance between you and \(E\), stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as \(A, B, C, D\), and \(E\) as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to \(B (30\text{ m}, -20\text{ m})\), \(A (60\text{ m}, 80\text{ m})\), \(E (-10\text{ m}, -10\text{ m})\), \(C (40\text{ m}, -30\text{ m})\), and \(D (-70\text{ m}, 60\text{ m})\)? State reasoning to show that the answer does not depend on the order in which the trees are labeled.