The nine Ring Wraiths want to fly from Barad-Dur to Rivendell. Rivendell is directly north of Barad-Dur. The Dark Tower reports that the wind is coming from the west at 52 miles per hour. In order to travel in a straight line, the Ring Wraiths decide to head northwest. At what speed should they fly (omit units)?

I drew it out but got lost from there, help would be nice.

Let the speed of the Ring Wraiths be x miles/hr. Since they're heading northwest, we can break this down into north and west components.

The north component will be the desired speed to travel directly towards Rivendell, and the west component will be to counteract the wind to keep their path straight.

By working with right triangles, observe that the north component will be x * cos(45) which simplifies to (x * √2)/2 since cos(45) = 1/√2. The west component will be x * sin(45), which is also (x * √2)/2 since sin(45) = 1/√2.

Since the wind is blowing from the west, to counteract it, the west component should equal the wind speed (52 mph). So we have:

(x * √2)/2 = 52

Now solve for x:

x * √2 = 104
x = 104 / √2
x ≈ 73.53 miles/hr

So the Ring Wraiths should fly at approximately 73.53 miles/hr.

To find the speed at which the Ring Wraiths should fly, we can use vector addition and the concept of resultant velocity.

Let's break down the given information:

- The wind is coming from the west at 52 miles per hour.
- The Ring Wraiths want to fly northwest in a straight line from Barad-Dur to Rivendell.

Now, let's establish a coordinate system, where the positive x-direction is east and the positive y-direction is north. Since Rivendell is directly north of Barad-Dur, the wind blowing from the west will have an effect on their flight path when they head northwest.

Given that the wind is coming from the west and towards the north, it can be resolved into its x and y components. The x-component of the wind will oppose the motion of the Ring Wraiths towards the northwest, while the y-component will not have any effect on their northward direction.

Let's calculate the x-component of the wind:

x-component = wind speed * cos(angle between wind and x-axis)

The angle between the wind and the x-axis can be found by taking the inverse tangent of the ratio of the y-component of the wind to the x-component of the wind:

Angle = arctan(y-component of the wind / x-component of the wind)

Now, let's calculate the x-component of the wind using the given speed of 52 miles per hour:

x-component = 52 mph * cos(Angle)

Next, we need to calculate the resultant velocity of the Ring Wraiths in the x-direction. Since they want to head northwest, the x-component of their velocity will be positive. We can represent their x-component as follows:

x-component of velocity = speed of the Ring Wraiths * cos(45 degrees)

Now, we can set up a vector equation to represent the forces acting in the x-direction:

Resultant force in the x-direction = x-component of velocity - x-component of the wind

Since the Ring Wraiths want to travel in a straight line, the resultant force in the x-direction should be zero. Therefore, we can set up the following equation:

0 = speed of the Ring Wraiths * cos(45 degrees) - x-component of the wind

Now, we can solve this equation to find the speed at which the Ring Wraiths should fly:

speed of the Ring Wraiths = x-component of the wind / cos(45 degrees)

Substituting the previously calculated values, we have:

speed of the Ring Wraiths = (52 mph * cos(Angle)) / cos(45 degrees)

Remember to convert all angles to radians before using trigonometric functions.

Calculating the specific values requires knowing the y-component of the wind. If you have that information, you can substitute it into the formula to find the required speed for the Ring Wraiths.

To find the speed at which the Ring Wraiths should fly, we can break down the situation into its components and use vector addition.

1. The wind speed is given as 52 miles per hour, coming from the west.
2. The Ring Wraiths plan to fly northwest, which means they are traveling in a direction between north and west.

To determine the speed at which the Ring Wraiths should fly, we need to find the resulting velocity that combines their desired northwest direction and the effect of the wind coming from the west.

Let's assume:
- The Ring Wraiths' desired speed is represented by v (unknown).
- The angle between the northwest direction and the northern direction is represented by θ (unknown).

To determine θ, we can use trigonometry. In a right-angled triangle, the angle θ is opposite to the side representing the desired northwest direction, and adjacent to the side representing the northern direction.

Now, let's break down the given wind speed vector into its north and west components:
- The west component of the wind speed can be calculated as 52 * cos(90°) = 0, as the wind is coming directly from the west.
- The north component of the wind speed can be calculated as 52 * sin(90°) = 52, as the wind is blowing directly north.

To combine the desired northwest direction with the wind's effect, we need to add the north components and west components separately.

1. For the north components: The Ring Wraiths want to fly directly north, and the wind is blowing directly north. Thus, the north component will be the sum of the desired velocity (v) and the wind's north component (52).

2. For the west components: The Ring Wraiths don't want the wind to affect their westward progression, so the west component will be the difference between their desired velocity (v) and the wind's west component (0).

Now we have two components represented as:
- North component: v + 52
- West component: v - 0 = v

To find the direction they are heading, we can use the tangent function of θ, which is the ratio of the north component to the west component:
tan(θ) = (v + 52) / v

Finally, to determine the speed the Ring Wraiths should fly, we want to find v. We can solve the equation above for v by isolating it:

tan(θ) = (v + 52) / v

Cross-multiplying:

v * tan(θ) = v + 52

Expanding:

v * tan(θ) - v = 52

Factoring out v:

v * (tan(θ) - 1) = 52

Dividing both sides by (tan(θ) - 1):

v = 52 / (tan(θ) - 1)

Now that we have the equation v = 52 / (tan(θ) - 1), we need the value of θ (the angle between north and northwest) to calculate the actual speed at which the Ring Wraiths should fly.