If a wheel has a nail stuck in the tire, the tire is 60m in diameter and the wheel drove R -234(1,0) how far did the nail travel?
To calculate how far the nail traveled, we need to determine the circumference of the tire and find the arc length of the part where the nail traveled.
First, let's find the circumference of the tire:
Circumference = π * diameter
Given that the diameter of the tire is 60 meters, we can calculate:
Circumference = π * 60 m
Circumference ≈ 188.5 m
Now we need to find the angle (in radians) that the wheel turned from the starting position to the position where the nail is located. The position is given as (1,0) on the unit circle. We can use the distance formula to find this angle:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (0,0) as the center of the unit circle and (1,0) as the nail's position, we have:
Distance = √((1 - 0)^2 + (0 - 0)^2)
Distance = √(1^2)
Distance = 1
Since the distance from the starting position to where the nail is located is equal to the radius of the unit circle (which is 1), the angle formed is equal to 1 radian.
Now, to find the arc length that the nail traveled:
Arc Length = Circumference * (angle in radians) / (2π)
Using the values we've calculated:
Arc Length = 188.5 m * (1 radian) / (2π)
Arc Length ≈ 30.02 m
Therefore, the nail traveled approximately 30.02 meters.