Final answer:
To determine the total displacement from the starting point and the direction to reach the treasure, we break down the distances and directions into x and y components. We then calculate the sum of the x and y components to find the total displacement and use trigonometry to determine the direction. The total displacement is approximately 42.826 paces and the direction is 116.13 degrees.
Step-by-step explanation:
To find the answer, we will break down the given distances and directions into x and y components. Let's start with the first leg of 18 paces at 45 degrees. The x component is 18 * cos(45) = 12.728 paces (westward) and the y component is 18 * sin(45) = 12.728 paces (northward). Next, for the second leg of 32 paces at 110 degrees, the x component is 32 * cos(110) = -10.386 paces (eastward) and the y component is 32 * sin(110) = 27.469 paces (northward). Lastly, for the third leg of 28 paces at 200 degrees, the x component is 28 * cos(200) = -21.180 paces (eastward) and the y component is 28 * sin(200) = -1.483 paces (southward).
To find the total displacement, we sum up the x and y components. The total x component is 12.728 - 10.386 - 21.180 = -18.838 paces and the total y component is 12.728 + 27.469 - 1.483 = 38.714 paces. Using the Pythagorean theorem, the distance from the starting point is sqrt((-18.838)^2 + 38.714^2) = 42.826 paces. To find the direction, we can use trigonometry. The angle can be determined using the inverse tangent function atan2(38.714, -18.838) = 116.13 degrees.