A circular walking track has a radius of 150 yards. Margaret walked 2/3 of the distance before taking a break.?
Would the path she walked be considered a “chord” or an “arc” of the circle? Explain.
– How many yards has Margaret walked? Show or explain how you got your answer.
– Margaret’s friend walked a path that resulted in her walking 400 yards. What was the central angle associated with a walk related to this distance. Show or explain how you got your answer.
If she walked on the track, it's an arc of a circle.
she walked 2/3 of the circumference. As you recall, the circumference of a circle of radius r is C = 2πr.
An arc of a circle of radius r that covers an angle θ is s = rθ
So, the friend walked through an angle θ such that
150θ = 400
To determine whether the path Margaret walked is considered a "chord" or an "arc" of the circle, we need to understand the definitions of these terms.
A "chord" is a line segment that connects two points on a circle's circumference. It can be any straight line within the circle.
An "arc" is a portion of the circle's circumference. It is a curved line that connects two points on the circle.
In Margaret's case, she walked a portion of the circular walking track. This means that she followed the track's curved path, connecting two points on the circle. Therefore, the path she walked would be considered an "arc" of the circle.
To calculate how many yards Margaret walked, we need to find 2/3 of the track's circumference. The formula for the circumference of a circle is given by C = 2πr, where C represents the circumference and r represents the radius.
Given that the radius is 150 yards, we can substitute this value into the formula:
C = 2π(150)
C = 300π
Now, to find 2/3 of the circumference, we multiply it by 2/3:
(2/3)(300π) ≈ 200π
Since we don't have an exact value for pi, we can approximate it as 3.14:
200π ≈ 200(3.14) ≈ 628 yards
Therefore, Margaret walked approximately 628 yards.
To find the central angle associated with a walk of 400 yards by Margaret's friend, we can use the concept of arc length and central angle relationship.
The formula to calculate the central angle θ in radians (where the arc length is given) is:
θ = arc length / radius
We are given that the arc length is 400 yards. Since the radius is still 150 yards, we can substitute these values into the formula:
θ = 400 / 150
= 8/3 radians
To convert this to degrees, we multiply by 180/π:
θ = (8/3)(180/π)
≈ 153.06 degrees
Therefore, the central angle associated with Margaret's friend walking a path of 400 yards is approximately 153.06 degrees.