At a county fair, a boy and two friends bring their teddy bears on the giant Ferris wheel. The wheel has a diameter of 14.8 m, the bottom of the wheel is 1.8 m above the ground and its rim is moving at a speed of 1.0 m/s. The boys are seated in positions 45° from each other. When the wheel brings the second boy to the maximum height, they all drop their stuffed animals. How far apart will the three teddy bears land? (Assume that the boy on his way down drops bear 1, and the boy on his way up drops bear 3.)
To solve this problem, we need to break it down into smaller steps.
Step 1: Find the distance traveled by the teddy bears on the Ferris wheel.
To find the distance traveled by the teddy bears, we need to calculate the circumference of the Ferris wheel. The formula for the circumference is C = πd, where C is the circumference and d is the diameter.
Given that the diameter of the wheel is 14.8 m, we can calculate the circumference:
C = π x 14.8 m = 46.55 m (rounded to two decimal places)
Step 2: Find the time it takes for the teddy bears to reach the top of the wheel.
The speed of the wheel is given as 1.0 m/s. To find the time it takes for the teddy bears to reach the top, we can use the formula d = vt, where d is the distance, v is the velocity, and t is the time.
Given that the distance is half the circumference (since they start from the bottom and go to the top), we have:
d = 46.55 m / 2 = 23.275 m
Now, we can rearrange the formula to solve for time:
t = d / v = 23.275 m / 1.0 m/s = 23.275 s
Step 3: Find the horizontal distance traveled by the teddy bears during the time they are in the air.
While the teddy bears are in the air, they will continue to move horizontally at a constant speed. The distance traveled horizontally can be calculated using the formula d = vt, where d is the distance, v is the velocity, and t is the time.
Given that the velocity is 1.0 m/s and the time is 23.275 s, we have:
d = 1.0 m/s x 23.275 s = 23.275 m
Step 4: Find the angular distance between each teddy bear.
The three teddy bears are seated 45° apart from each other. Since the wheel completes one full revolution (360°) during the time it takes for the teddy bears to reach the top, we can calculate the angular distance covered by each teddy bear.
The angular distance covered by each teddy bear can be calculated as follows:
angular distance = (45° / 360°) x circumference
angular distance = (45° / 360°) x 46.55 m = 5.819 m
Step 5: Find the final distance between the teddy bears.
To find the final distance between the teddy bears, we need to add the horizontal distance traveled by each teddy bear to the angular distance between them.
The final distance between the teddy bears can be calculated as follows:
final distance = horizontal distance + angular distance
final distance = 23.275 m + 5.819 m = 29.094 m
Therefore, the three teddy bears will land approximately 29.094 meters apart.