The car in the figure below travels a distance of 745 m along the track, and the driver finds that she has traveled through an angle of 475°. What is the radius of the track?
To find the radius of the track, we can use the relationship between the distance traveled, the angle covered, and the radius of the track.
The distance traveled along the track is 745 m, and the angle covered is 475°. We know that in a circle, the circumference (distance around the circle) is equal to 2π times the radius.
First, we need to convert the angle from degrees to radians because the formula for circumference uses radians.
To convert degrees to radians, we use the formula:
radians = degrees * π / 180.
So, the angle in radians is:
475° * π / 180.
Now, we can calculate the circumference of the circle using the formula:
circumference = 2π * radius.
Since we are given the distance traveled along the track, which is equal to the circumference of the circle, we can equate the two equations:
745 m = 2π * radius.
We can now solve for the radius by dividing both sides of the equation by 2π:
radius = 745 m / (2π).
Calculating this, we get:
radius ≈ 118.53 m.
Therefore, the radius of the track is approximately 118.53 meters.