Two persons each hold the end of a 20 ft long jumping rope. How far apart should they stand if they hold the rope 4.5 ft off the ground, and the middle of the rope just touches the ground? Assume the arc formed by the jumping rope is circular.
I know that this problem involves an arc and a segment, but I can't find a formula to help me. The answer is 17, but I don't know how to get there.
Draw a diagram. If the circle has radius r, and the girls stand a distance 2x apart, and the rope subtends an angle 2θ, we have
x^2 + (r-4.5)^2 = r^2
x/(r-4.5) = sinθ
r sin2θ = 20
If you wade through that, you will get x ≈ 17
sinè = x/r and 20 = rè, but thanks anyway. you got me thinking on the right track.
To solve this problem, we can use the properties of circles. Let's break it down step by step:
1. Draw a diagram: Begin by drawing a circle representing the arc formed by the jumping rope. Mark the center of the circle as point O, and draw a horizontal line from point O to represent the ground.
2. Add the known information: Mark two points on the circle to represent where the two persons are holding the rope. Label these points as A and B. Additionally, draw a vertical line from the center of the circle to the midpoint of the rope, as it touches the ground. Label this point as M.
3. Define the variables: Let the distance between points A and B (the two persons) be x. We need to find the value of x.
4. Use the properties of a circle: In this case, we can observe that triangles OAM and OBM are similar, as they share a common angle at O. Therefore, we can write:
OA / OB = AM / BM
Given that the length of the rope is 20 ft and it touches the ground at point M, we have:
OA + AM = OB + BM = 20 ft
Since AM and BM are equal, we can denote them as a.
5. Solve the equations: Using the information above, we can set up two equations:
OA + a = 20
OB + a = 20
Rearranging these equations, we get:
OA = 20 - a
OB = 20 - a
Substituting these values into the similarity equation above, we have:
(20 - a) / (20 - a) = a / a
Simplifying this equation, we get:
20 - a = a
Solving for a, we find:
2a = 20
a = 10 ft
6. Calculate the distance between the two persons: Now that we have the value of a, which represents the distance from each person to the middle of the rope, we can find the total distance between the two persons by adding twice the value of a:
x = 2 * a
x = 2 * 10 ft
x = 20 ft
Therefore, the two persons should stand 20 feet apart to hold the rope at 4.5 feet off the ground, with the middle of the rope just touching the ground.
I hope this explanation helps!