A ship leaves the island of Guam and sails a distance 280 at an angle 38.0 north of west.In which direction must it now head so that its resultant displacement will be 115 directly east of Guam? (Express your answer as an angle measured south of east)

How far must it sail so that its resultant displacement will be 115 directly east of Guam?God bless you

To find the direction the ship must head in order for its resultant displacement to be 115 miles directly east of Guam, we must consider the vectors involved.

Let's break down the information provided:

1. The ship sails a distance of 280 miles at an angle of 38.0° north of west. This can be represented as a vector with a magnitude of 280 miles and an angle of 38.0° north of west.

2. The ship wants its resultant displacement to be 115 miles directly east of Guam. This can be represented as a vector with a magnitude of 115 miles and a direction of directly east, which is 0°.

To find the direction the ship must now head, we need to find the angle between the resultant vector and the east direction. This can be calculated by subtracting the angle of the original vector from the desired east direction angle.

So, the angle between the resultant vector and the east direction is:

= 0° - 38.0°
= -38.0°

Since the angle calculated is measured clockwise from the east direction, we need to express the answer as an angle measured counterclockwise from the east direction.

Counterclockwise angles are typically referred to as positive angles, so we can convert -38.0° to a positive angle by adding 360°:

= -38.0° + 360°
= 322.0°

Therefore, the ship must head in a direction 322.0° counterclockwise from the east direction.

To find how far the ship must now sail, we can use the Pythagorean theorem since the resultant displacement is the hypotenuse of a right triangle formed by the original vector and the desired eastward displacement.

The magnitude of the original vector is given as 280 miles, and the magnitude of the desired eastward displacement is given as 115 miles.

Let's call the distance the ship must sail as "x". Applying the Pythagorean theorem:

x^2 = 115^2 + 280^2
x^2 = 13225 + 78400
x^2 = 91625

Taking the square root of both sides:

x ≈ √91625
x ≈ 302.7

Therefore, the ship must sail approximately 302.7 miles to achieve a resultant displacement of 115 miles directly east of Guam.

To find the direction in which the ship must head to achieve a resultant displacement of 115 miles directly east of Guam, we need to find the angle between the initial displacement and the resultant displacement.

Given:
Initial displacement = 280 miles
Initial angle (north of west) = 38 degrees
Resultant displacement = 115 miles directly east

To solve this problem, we can use vector addition. First, let's convert the initial displacement to rectangular form.

The initial displacement can be split into its northward and westward components using trigonometry. The northward component (y-component) is given by:

y-component = initial displacement * sin(angle)
y-component = 280 * sin(38)
y-component = 280 * 0.6157
y-component ≈ 172.59 miles (rounded to two decimal places)

The westward component (x-component) is given by:

x-component = initial displacement * cos(angle)
x-component = 280 * cos(38)
x-component = 280 * 0.7880
x-component ≈ 220.64 miles (rounded to two decimal places)

Now, let's find the resultant displacement using vector addition. Since the resulting displacement is 115 miles directly east, the y-component of the resultant displacement would be 0 (as there is no northward displacement).

To find the x-component of the resultant displacement, we can subtract the x-component of the initial displacement from the desired eastward displacement:

x-component resultant = desired eastward displacement - x-component initial
x-component resultant = 115 - 220.64
x-component resultant ≈ -105.64 miles (rounded to two decimal places)

Now, we can find the angle between the initial displacement and the resultant displacement.

angle = arctan(y-component / x-component resultant)
angle = arctan(172.59 / -105.64)
angle ≈ -57.89 degrees (rounded to two decimal places)

Since the question asks for the angle measured south of east, we need to subtract the obtained angle from 180 degrees to get the final answer:

angle measured south of east = 180 - 57.89
angle measured south of east ≈ 122.11 degrees (rounded to two decimal places)

Therefore, the ship must head in a direction approximately 122.11 degrees south of east to achieve a resultant displacement of 115 miles directly east of Guam.

To find how far it must sail to achieve this resultant displacement, we can use the Pythagorean theorem:

Resultant displacement^2 = x-component resultant^2 + y-component resultant^2
115^2 = (-105.64)^2 + 0^2
13225 = 11158.08 + 0
Distance = sqrt(13225 - 11158.08)
Distance ≈ 59.96 miles (rounded to two decimal places)

Therefore, the ship must sail approximately 59.96 miles to achieve a resultant displacement of 115 miles directly east of Guam.