Final answer:
To find the displacement of a rollercoaster after it moves horizontally and then at an angle, we calculate the horizontal and vertical components of the angled movement and then apply the Pythagorean theorem to find the rollercoaster's total displacement. The displacement is approximately 126.05 m.
Step-by-step explanation:
The question asks to calculate the displacement of a rollercoaster from its starting point after moving horizontally and then at an angle above the horizontal. To solve this, we need to break the movement at the angle into horizontal and vertical components and then add them to the initial horizontal movement.
The rollercoaster first moves 85 m horizontally, then travels 45 m at a 30° angle above the horizontal. To find the total horizontal displacement, we add the horizontal component of the angled movement to the initial 85 m. The horizontal component can be found using trigonometry, specifically the cosine function:
Horizontal component = 45 m * cos(30°)
Similarly, the vertical displacement is found using the sine function:
Vertical component = 45 m * sin(30°)
We then use the Pythagorean theorem to find the total displacement, which is the hypotenuse of the right-angled triangle formed by the horizontal and vertical components:
Total displacement = √(horizontal displacement² + vertical displacement²)
Calculating the components:
- Horizontal component = 45 m * cos(30°) = 45 m * (√3/2) = 38.97 m (approximately)
- Vertical component = 45 m * sin(30°) = 45 m * (1/2) = 22.5 m
Total horizontal displacement = 85 m + 38.97 m = 123.97 m
Total displacement = √(123.97 m² + 22.5 m²) = √(15382.65 m² + 506.25 m²) = √(15888.9 m²) = 126.05 m (approximately)
The rollercoaster's displacement from its starting point is approximately 126.05 m.