Final answer:
After the Wonder Wheel Ferris wheel has rotated 202.5° counterclockwise, the car is approximately 41.0236 feet above the ground. This is calculated by using the cosine of 67.5° to find the vertical component of the radius and then adding the base height of 15 feet.
Step-by-step explanation:
To calculate how high above the ground the car on the Wonder Wheel Ferris wheel at Coney Island is after rotating 202.5° counterclockwise, we can use the concepts of circular motion and trigonometry. The Ferris wheel has a radius of 68 feet, and the base of the Ferris wheel is 15 feet above the ground. When the Ferris wheel rotates 202.5°, the car will be at a position between the very top and the horizontal to the left of the vertical diameter.
We can visualize this situation as the car being at the endpoint of the radius that forms a 202.5° angle with the vertical diameter line (where 180° would be exactly horizontal). To find the vertical component of this radius, we can use the cosine function, as the cosine of the angle will give us the proportion of the radius that is in line with the vertical diameter from the center of the wheel:
Cos(67.5°) = vertical component / radius
Plugging in the radius, we get:
Cos(67.5°) = vertical component / 68 feet
vertical component = Cos(67.5°) × 68 feet
To find how high the car is above the ground, we then add the base height of the wheel plus the vertical component:
height above the ground = base height + vertical component
If we do the calculations:
vertical component ≈ 0.3827 × 68 feet = 26.0236 feet
height above the ground = 15 feet + 26.0236 feet ≈ 41.0236 feet
So, the car is approximately 41.0236 feet above the ground when the Ferris wheel stops.