a roller coaster car moves 200 ft horizontally and then rises 135 ft at an angle of 30 degrees above the horizontal. It then travels 135 ft at an angle of 40 degrees downward. what is its displacement from its starting point?

im not really sure what to do, I've drawn a graph but then got stuck.

Thanks!!!

I am not sure either because you did not say if the track circled around or went straight, for example East, the whole time.

I will have to assume it went straight.
In that case we will have to figure out how far East it went and how far up it went, then see how far that point is from the starting point
East 200
East 135 cos 30 , up 135 sin 30
East 135 cos 40 , up -135 sin 40
so East
200 + 117 + 103 = 420
and up
67.5 - 86.8 = -19.3
so the hypotenuse is sqrt (420^2 +19.3^2)

To solve this problem, we can break down the motion of the roller coaster car into its horizontal and vertical components.

1. Horizontal motion: The car moves 200 ft horizontally, which means its horizontal displacement is 200 ft.

2. Vertical motion: The car rises 135 ft at an angle of 30 degrees above the horizontal, and then travels 135 ft at an angle of 40 degrees downward.

To determine the vertical displacement, we need to find the vertical components of both motions. We can use trigonometry to do this:

- For the first motion (rising 135 ft at an angle of 30 degrees above the horizontal), we can find the vertical component by multiplying the total displacement by the sine of the angle: 135 ft * sin(30 degrees) = 67.5 ft.

- For the second motion (traveling 135 ft at an angle of 40 degrees downward), we need to consider that it is downward, so we'll use the negative value of the vertical component. The vertical component is given by: -135 ft * sin(40 degrees) = -87.1122 ft (rounded to four decimal places).

Now, we can calculate the total vertical displacement by adding the vertical components of both motions: 67.5 ft + (-87.1122 ft) = -19.6122 ft (rounded to four decimal places).

To determine the displacement from the starting point, we need to calculate the magnitude of the resultant displacement. We can use the Pythagorean theorem to find the magnitude of the displacement:

Displacement = √(Horizontal displacement^2 + Vertical displacement^2)
Displacement = √(200^2 + (-19.6122)^2) = √(40,000 + 384.5968) = √40,384.5968

Calculating the square root (approximated to four decimal places), we get:
Displacement = 201.9582 ft

Therefore, the displacement of the roller coaster car from its starting point is approximately 201.9582 ft.

To solve this problem, we can break down the motion of the roller coaster car into two separate parts: the horizontal motion and the vertical motion.

1. Horizontal Motion:
The car moves 200 ft horizontally, which means there is no change in the vertical position. Therefore, the horizontal displacement is 200 ft.

2. Vertical Motion:
The car first rises 135 ft at an angle of 30 degrees above the horizontal. To calculate the vertical displacement, we can use trigonometry. The vertical displacement can be calculated as follows:
Vertical displacement = vertical distance * sin(angle)
Vertical displacement = 135 ft * sin(30 degrees)
Vertical displacement = 135 ft * 0.5
Vertical displacement = 67.5 ft (rounded to one decimal place)

Next, the car travels 135 ft at an angle of 40 degrees downward. To calculate the vertical displacement, we can again use trigonometry:
Vertical displacement = vertical distance * sin(angle)
Vertical displacement = 135 ft * sin(40 degrees)
Vertical displacement = 135 ft * 0.6428
Vertical displacement = 86.85 ft (rounded to two decimal places)

Now, let's find the net vertical displacement. Since the car rises first and then descends, we need to consider the directions:
Net vertical displacement = vertical displacement (rising) - vertical displacement (descending)
Net vertical displacement = 67.5 ft - 86.85 ft
Net vertical displacement = -19.35 ft (rounded to two decimal places)

Finally, we can find the overall displacement of the car from its starting point using the Pythagorean theorem:
Overall displacement = sqrt(horizontal displacement^2 + net vertical displacement^2)
Overall displacement = sqrt((200 ft)^2 + (-19.35 ft)^2)
Overall displacement = sqrt(40000 ft^2 + 374.8225 ft^2)
Overall displacement = sqrt(40374.8225 ft^2)
Overall displacement ≈ 201.9 ft (rounded to one decimal place)

Therefore, the displacement of the roller coaster car from its starting point is approximately 201.9 ft.