A town y is 200km from town x in a direction 040 degree. How far is y east of x?

128.6km

A town of Y is 200kilometer from X in a direction 040. How far is Y east of X. Please show with working. Thanks

It is convenient to use the Cartesian coordinates with x due east, and y due north.

Then 040 (bearing) is N40°E, which in the Cartesian system is θ=90-40=50° counterclockwise from the x-axis.
Then we apply the standard formulae:
δx=rcos(θ) (due east positive)
δy=rsin(θ) (due north positive).
where r is the distance = 200 km.

can another method be used pls?

Anthony show me the workings

Well, if town Y is located 200km from town X in a direction of 040 degrees, we can break out our trusty compass and take a chuckle at this situation.

Now, 040 degrees implies that we are heading northeast. So, if we were to draw a straight line between town X and town Y, we would notice that town Y is indeed northeast of town X.

However, when it comes to the distance east, well, we have a bit of a twist here! Since we are heading northeast, it means that we are not traveling purely east. So, we can't say that town Y is directly east of town X.

To determine how far town Y is east of town X, we can use some trigonometric tomfoolery. By using the cosine function, we can find the eastward displacement. In this scenario, the cosine of 040 degrees would give us the answer we seek.

But hey, who needs all those calculations when we can just say that town Y is, well, northeast of town X? It saves us the trouble of breaking out the protractors and trigonometry textbooks!

To determine how far town Y is east of town X, we need to break down the given information.

The direction "040 degrees" refers to a compass bearing. In the context of compass bearings, 0 degrees points due North, 90 degrees points due East, 180 degrees points due South, and 270 degrees points due West.

In this case, a direction of 040 degrees means it is slightly northeast of east (since 40 degrees is closer to east than north).

Now, let's find out the vertical and horizontal components of the distance between the two towns using basic trigonometry.

The vertical component represents the North-South distance, and the horizontal component represents the East-West distance.

Given that the total distance between the towns is 200 km, the vertical distance is given by:
Vertical distance = Total distance * cosine of the angle
Vertical distance = 200 km * cos(40 degrees)

Now, let's calculate the vertical distance:
Vertical distance = 200 km * cos(40 degrees) ≈ 153.53 km (rounded to two decimal places)

Since we are interested in how far Town Y is east of Town X, we need to find the horizontal distance.

Horizontal distance = Total distance * sine of the angle
Horizontal distance = 200 km * sin(40 degrees)

Now, let's calculate the horizontal distance:
Horizontal distance = 200 km * sin(40 degrees) ≈ 128.18 km (rounded to two decimal places)

Therefore, Town Y is approximately 128.18 km east of Town X.