A ship leaves the island of Guam and sails a distance 280 at an angle 39.0 north of west.
1.In which direction must it now head so that its resultant displacement will be 125 directly east of Guam? (Express your answer as an angle measured south of east)
2.How far must it sail so that its resultant displacement will be 125 directly east of Guam?
To solve these questions, we can use vector addition and trigonometry. Let's break it down step by step:
1. In which direction must it now head so that its resultant displacement will be 125 directly east of Guam?
To determine the direction, we need to find the angle between the resultant displacement and the east direction. Here's how we can proceed:
a. We know that the ship initially sailed for a distance of 280 units at an angle of 39.0° north of west. Since it's north of west, we can subtract the angle from 90° to get the angle relative to the north direction.
Therefore, the angle relative to the north is 90° - 39.0° = 51.0°.
b. We want the resultant displacement to be directly east, which means the angle between the east direction and the resultant displacement is 90°. We'll subtract this angle from 90° to find the angle south of east.
Therefore, the angle required is 90° - 51.0° = 39.0° south of east.
So, the ship must now head in a direction 39.0° south of east to achieve a resultant displacement of 125 directly east of Guam.
2. How far must it sail so that its resultant displacement will be 125 directly east of Guam?
To find the distance it must sail, we can use Pythagoras' theorem because the displacement vectors form a right triangle.
a. We have the length of the resultant displacement, which is 125 units (directly east).
b. To find the lengths of the other two sides (lets call them x and y), we need to use trigonometry. We have the angle 51.0° and we know that the opposite side (y) corresponds to the distance of 280 units.
Using trigonometry, we can determine the length of the adjacent side (x) relative to the angle of 51.0°:
x = 280 * cos(51.0°)
c. Now, we can use Pythagoras' theorem to find the length of the remaining side (which corresponds to the distance the ship must sail):
(distance)^2 = x^2 + y^2
(distance)^2 = (280 * cos(51.0°))^2 + (280 * sin(51.0°))^2
Taking the square root of both sides will give us the distance:
distance = sqrt((280 * cos(51.0°))^2 + (280 * sin(51.0°))^2)
Calculating this expression will give you the distance the ship must sail to achieve a resultant displacement of 125 units directly east of Guam.