SOLUTION: You are a lifeguard at a pong. The pond is a circle with a diameter of 600 feet. You want to rope off a section of the pond for swimming. If you want the rope to form a chord of th

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: You are a lifeguard at a pong. The pond is a circle with a diameter of 600 feet. You want to rope off a section of the pond for swimming. If you want the rope to form a chord of th      Log On


   

Question 721507: You are a lifeguard at a pong. The pond is a circle with a diameter of 600 feet. You want to rope off a section of the pond for swimming. If you want the rope to form a chord of the circle and have a maximum distance of 120 feet from shore, approximately how long will you need the rope to be?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
It will probably help if you have a diagram:
  1. Draw a circle centered at the origin of a graph.
  2. Since the diameter is 600 feet, the radius is 300 feet. The points of the circle on the axes will be:
    (0 + 300, 0) or (300, 0)
    (0 - 300, 0) or (-300, 0)
    (0, 0 + 300) or (0, 300)
    (0, 0 - 300) or (0, -300)
  3. The points on the circle are where the shore is. Let's choose (300, 0) as the point on the shore from which we will measure the distance to the rope. 120 feet into the lake (inside the circle) from (300, 0) would be (300 - 120, 0) or (180, 0)
  4. Draw a vertical line through (180, 0). This vertical line will intersect the circle in two places. Let's call them A and B. Segment AB represents the rope.
I hope the diagram makes sense to you. Our problem is to find the length of the rope, or the length of segment AB. The way we will do this is
  1. Find the equation of the circle. Our circle has a center of (0, 0) and a radius of 300. So its equation will be:
    %28x-0%29%5E2%2B%28y-0%29%5E2=%28300%29%5E2
    or
    x%5E2%2By%5E2=90000
  2. Use the equation and the fact that the x-coordinates of A and B are 180 to find the y-coordinates of A and B. %28180%29%5E2%2By%5E2=90000
    32400%2By%5E2=90000
    y%5E2+=+57600
    y = +240
  3. Use the y-coordinates to find the length of the rope. Since A and B are on the same vertical line, the distance between them is simply the difference in the y-coordinates. (If you don't believe me, use the full distance formula.) So the length of the rope will be:
    240-(-240) = 480 feet.