You and a group of friends are participating in a ropes course. One of the obstacles involves doing a Tarzan Swing into a lake. A rope swing is at the edge of the lake. It is attached to a platform arm that overhangs the lake as shown in the diagram below. The point at which the rope is attached is 30 feet above the water and 20 feet from the bank. The platform is on the 20-foot bank at the edge of the lake and is 20 feet high. The rope is 35 feet long, allowing the rope to hang down in the water so that a swimmer can reach it and tow it back to shore or the next group member. The rope is knotted every five feet so that swingers have something to hold as they swing out over the lake. How long is the part of the rope that is stretched from the attachment point to the edge of the bank?
To determine the length of the rope that is stretched from the attachment point to the edge of the bank, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle formed by the rope, the bank, and the height of the platform. One leg of the triangle is the distance from the attachment point to the bank, which is given as 20 feet. The other leg is the height of the platform, also given as 20 feet. The hypotenuse is the length of the rope, which is 35 feet.
Let's use the Pythagorean theorem to find the length of the stretched rope:
hypotenuse^2 = leg1^2 + leg2^2
35^2 = 20^2 + 20^2
1225 = 400 + 400
1225 = 800
1225 - 800 = 425
The length of the part of the rope that is stretched from the attachment point to the edge of the bank is the square root of 425, which is approximately 20.62 feet.