A roller coaster moves 200 ft horizontally and then rises 135 ft at an angle of 30 degrees above the horizontal. Next, it travels 135 ft at an angle of 40 degrees below the horizontal. Find the roller coasters displacement from its starting point to the end of this movement. Answer = 421 ft at 3 degrees below the horizontal. Please show work on how to solve for this thank you.

To find the roller coaster's displacement, we need to break down its movement into its horizontal and vertical components. Let's start by finding the horizontal displacement and vertical displacement separately.

1. Horizontal Displacement:
The roller coaster moves 200 ft horizontally, so the horizontal displacement is simply 200 ft.

2. Vertical Displacement:
The roller coaster rises 135 ft at an angle of 30 degrees above the horizontal. To find the vertical displacement, we need to calculate the vertical component of this movement.

Using trigonometry, we can find the vertical component as follows:
Vertical Component = (Rise) * sin(angle)
Vertical Component = 135 ft * sin(30 degrees)
Vertical Component = 135 ft * 0.5
Vertical Component = 67.5 ft

Now, the roller coaster travels 135 ft at an angle of 40 degrees below the horizontal. This means it goes down in height. To find the vertical component, we'll use the same formula as above but with a negative sign.

Vertical Component = (Rise) * sin(-angle)
Vertical Component = 135 ft * sin(-40 degrees)
Vertical Component = 135 ft * (-0.6428)
Vertical Component = -86.78 ft

3. Calculate the Total Vertical Displacement:
To find the total vertical displacement, we need to sum up the vertical components of both movements:

Total Vertical Displacement = (Vertical Component of 30 degrees) + (Vertical Component of -40 degrees)
Total Vertical Displacement = 67.5 ft + (-86.78 ft)
Total Vertical Displacement = -19.28 ft

Now that we have the horizontal and vertical displacements, we can calculate the total displacement.

Total Displacement = sqrt((Horizontal Displacement)^2 + (Vertical Displacement)^2)
Total Displacement = sqrt((200 ft)^2 + (-19.28 ft)^2)
Total Displacement ≈ 201.05 ft

Additionally, the angle of the total displacement below the horizontal can be found using trigonometry:
Angle = atan((Vertical Displacement) / (Horizontal Displacement))
Angle = atan(-19.28 ft / 200 ft)
Angle ≈ -3.21 degrees

Therefore, the roller coaster's displacement from its starting point to the end of this movement is approximately 201.05 ft at an angle of -3.21 degrees below the horizontal.

To solve this problem, we can break down the movement of the roller coaster into horizontal and vertical components. Using trigonometry, we can find the horizontal and vertical displacements separately and then combine them to find the overall displacement.

First, let's find the horizontal displacement:
The roller coaster moves 200 ft horizontally, so the horizontal displacement is 200 ft.

Next, let's find the vertical displacements:
The roller coaster rises 135 ft at an angle of 30 degrees above the horizontal. To find the vertical displacement, we can use the sine function:
Vertical displacement = 135 ft * sin(30°) = 135 ft * 0.5 = 67.5 ft (upwards)

The roller coaster then travels 135 ft at an angle of 40 degrees below the horizontal. To find the vertical displacement, we can again use the sine function:
Vertical displacement = 135 ft * sin(40°) = 135 ft * (-0.6428) = -86.85 ft (downwards)

Now, let's calculate the overall displacement:
The overall horizontal displacement is 200 ft, and the overall vertical displacement is 67.5 ft upwards and 86.85 ft downwards.
To combine the horizontal and vertical displacements, we can use the Pythagorean theorem.

Overall displacement = √(Horizontal displacement^2 + Vertical displacement^2)
Overall displacement = √(200^2 + (67.5 - 86.85)^2)
Overall displacement = √(40,000 + 3620.3025)
Overall displacement = √43,620.3025
Overall displacement = 209.057 ft

Next, we need to determine the angle of this overall displacement.
We can use the inverse tangent function:
Angle = tan^(-1)(Vertical displacement / Horizontal displacement)
Angle = tan^(-1)((67.5 - 86.85) / 200)
Angle = tan^(-1)(-0.09675)
Angle = -3.014 degrees

Finally, the roller coaster's displacement from its starting point to the end of this movement is approximately 209.057 ft at an angle of 3 degrees below the horizontal.

hhf