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.

Question

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.

Answer 1

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

Answer 2

sinè = x/r and 20 = rè, but thanks anyway. you got me thinking on the right track.