Question 721507
It will probably help if you have a diagram:<ol><li>Draw a circle centered at the origin of a graph.</li><li>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)</li><li>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)</li><li>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.</li></ol>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<ol><li>Find the equation of the circle. Our circle has a center of (0, 0) and a radius of 300. So its equation will be:
{{{(x-0)^2+(y-0)^2=(300)^2}}}
or
{{{x^2+y^2=90000}}}</li><li>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. {{{(180)^2+y^2=90000}}}
{{{32400+y^2=90000}}}
{{{y^2 = 57600}}}
y = <u>+</u>240</li><li>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.</li></ol>