Instructions for finding a buried treasure include the following: Go 745.6 paces at 298◦, turn to 204◦ and walk 178 paces, then travel 490 paces at 55◦

Find the magnitude of the resultant displacement from the starting point. Answer in Units of paces.
(Part 2)
What is the direction of the resultant displacement? Use counterclockwise from due east as the positive angular direction, between the limits of −180◦ and +180◦. Answer in units of ◦

D = 745.6[298o] + 178[204o] + 490[55o]

X=745.6*cos298 + 178*cos204 + 490*cos55=
468.5 Paces.

Y=745.6*sin298 + 178*sin204 + 490*sin55=
-329.3 Paces.

Tan A = Y/X = -329.3/468.5 = -70291
A = -35.1o = 324.9o CCW = Direction.

Displacement=X/Cos A = 468.5/Cos 324.9 = 573 Paces.

Finding buried treasure, huh? That's exciting! Let's calculate the resultant displacement and its direction for you.

To find the magnitude of the resultant displacement, we'll use the Pythagorean theorem.

First, let's break down the given instructions:
- Go 745.6 paces at 298°
- Turn to 204° and walk 178 paces
- Travel 490 paces at 55°

Now, let's do some math!

For the first step, we'll use the given distance and angle to find the x and y components. Using trigonometry, the x-component is 745.6 * cos(298°) and the y-component is 745.6 * sin(298°).
For the second step, the x-component is 178 * cos(204°) and the y-component is 178 * sin(204°).
And finally, for the third step, the x-component is 490 * cos(55°) and the y-component is 490 * sin(55°).

Next, we'll add up all the x-components and y-components separately to find the final x and y coordinates.

Finally, the magnitude of the resultant displacement is found using the Pythagorean theorem: magnitude = sqrt(x^2 + y^2).

To find the direction of the resultant displacement, we'll use some trigonometry again. We'll calculate the angle of the displacement using the equation θ = tan⁻¹(y-component / x-component), and then adjust the angle to fall within the limits of -180° to +180°.

So grab your calculator, and let's get calculating!

To find the magnitude of the resultant displacement, we need to calculate the total distance traveled in both the x and y directions.

1. Convert the given distances and angles to components:
- For the first step, 745.6 paces at 298◦, we can break it down into:
- x-component: 745.6 cos(298◦)
- y-component: 745.6 sin(298◦)

- For the second step, 178 paces in the direction of 204◦, we can break it down into:
- x-component: 178 cos(204◦)
- y-component: 178 sin(204◦)

- For the third step, 490 paces at 55◦, we can break it down into:
- x-component: 490 cos(55◦)
- y-component: 490 sin(55◦)

2. Calculate the total x and y components by summing up the respective components from each step.

3. Calculate the magnitude of the resultant displacement using the Pythagorean theorem:
- Resultant magnitude = sqrt((total x component)^2 + (total y component)^2)

For the direction, we can calculate the angle between the positive x-axis and the resultant displacement vector using the arctan function:
- Resultant angle = arctan(total y component / total x component)

The answer to the magnitude of the resultant displacement will be in paces, and the answer to the direction will be in degrees (positive counterclockwise from due east).

Calculations can be performed using a calculator or a programming language.

To find the magnitude and direction of the resultant displacement, we need to use vector addition.

Step 1: Convert the given information into vector form.

The first instruction tells us to go 745.6 paces at 298°. Let's represent this as vector A:

A = 745.6 paces at 298°

The second instruction tells us to turn to 204° and walk 178 paces. Let's represent this as vector B:

B = 178 paces at 204°

The third instruction tells us to travel 490 paces at 55°. Let's represent this as vector C:

C = 490 paces at 55°

Step 2: Add the vectors together to find the resultant displacement.

To add the vectors, we need to break them down into their horizontal (x) and vertical (y) components. We can use trigonometry to do this.

For vector A:
Ax = 745.6 * cos(298°)
Ay = 745.6 * sin(298°)

For vector B:
Bx = 178 * cos(204°)
By = 178 * sin(204°)

For vector C:
Cx = 490 * cos(55°)
Cy = 490 * sin(55°)

Now, we can add the components together:

Rx = Ax + Bx + Cx
Ry = Ay + By + Cy

Step 3: Find the magnitude of the resultant displacement.

The magnitude of the resultant displacement can be found using the Pythagorean theorem:

R = sqrt(Rx^2 + Ry^2)

Calculate the value of R using the components from Step 2.

Step 4: Find the direction of the resultant displacement.

Use trigonometry to find the angle of the resultant displacement:

θ = atan(Ry / Rx)

Convert the result from radians to degrees.

This will give you both the magnitude and direction of the resultant displacement.