The Treasure of Pirate Island You find a treasure map showing where a pirate captain buried his treasure chest. The map shows the path the pirates took from where they landed to where they buried the chest, but it is not very accurate. However the map came with a letter: We made land at the beach on the Northwest side of the island. Starting there from the largest tree, we could see the smoke coming out of the volcano. We made directly for the volcano, stopping after walking two-fifths of the way. From there, we spied a hill with a large outcropping at its top. In the center of that, the sun shone through a hole in the rock like a window. We turned towards the window and kept walking until the rocks blocked the sun. This was about one-half of the way to the window, in the middle of a saddle. We could look down and see the lagoon on the Southeast side of the island from there, so we walked down towards it until the trees obscured the lagoon, about two-fifths, and buried the treasure there. You've done some additional research, and found the positions from the bottom left-hand corner for the four named places in the text: the pirates' beach A, the volcano B, the hill C, and the lagoon D: Beach A = 5800 ft E + 7100 ft N Volcano B = 14500 ft E + 5300 ft N Window C = 7000 ft E + 1900 ft N Lagoon D = 17300 ft E + 4300 ft N In this problem, you will determine where it is that X marks the spot. Part 1 of 3 - Half Way to B In this preliminary step, you will look at a direct interpretation of finding the midpoint of the line segment defined by two positions. That is: to find the point half way between A and B, you need to add half the displacement from A to B to the position A. M = X + 4 0-A) What is the displacement between A and B? B - A = 8700 ft / 8700 ft E + -1800 ft -1800 ft N Now, use that to find the midpoint M = 10350 ft / 10200 ft E + 5650 ft x / 6200 ft N Part 2 of 3 - Another Way to Line The following function describes a line between two points A and B f(a) = (1-a)* + aB where a is a unitless parameter. Below you will use the two points A (the beach) and B (the volcano) to explore this. It would be best if you were to draw out the locations on a piece of paper before working this. Find the value of f at the following positions: F(0) = 5800 ft / 5800 ft E + 7100 ft $ 7100 ft N (2/5) = 9180 ft / 9280 ft E + 6460 ft / 6380 ft N F(1/2) = 10350 ft / 10200 ft E + 6200 fe 6200 ft N f(1) = 14500 ft 14500 ft E + 5300 ft 5300 ft N When a = 0, f(0) = A, when a = 1, f = B. What does it mean when a = -1>?> The minus sign switches the variables, so f(-1) = -A. The minus sign reverses the result, so f(-1) = -B. It is a point on a line defined by A and B. The minus sign doesn't mean anything, so f(-1) = B. The minus sign switches the variables, so f(-1) = A.Part 3 of 3 - The Path to X Now, use the procedures in the last section to find the various points. [Note: even though I have already drawn out a qualitative map for you, it would be helpful for you to draw the map yourself using your own numbers.] Where do you stop walking toward the volcano? L = 9180 ft . E + 6460 ft N Where do you stop walking toward the window? N = 5250 ft x E + 1800 ft x N Where is the treasure? X = JE + Submit Forfeit Part