Boating: where should your passengers be when you are tied to the dock fueling your vessel?

A 20-ft-by-4-ft dock is anchored in the middle of a lake. An inner tube is tied to the bottom corner of the dock with a 13-ft rope?

• A 20-ft-by-4-ft dock is anchored in the middle of a lake. An inner tube is tied to the bottom corner of the dock with a 13-ft rope.find the area of the region in which the inner tube ...show more

• Answer:

  Since we don't know the diameter of the inflated inner tube, we can only make a low estimate of the travel area, which will be an arc around the corner of the dock. If you make a little sketch, you will see that the travel area comprises 3/4 of a 13' radius circle plus 1/4 of a 9' radius circle* This amounts to π[(3/4) x 169 + (1/4) x 81) = 147π ft² this is the area swept out by the attachment point on the inner tube, which will be nearest the dock. It ignores the added area covered by the outer side of the tube by treating the tube as a point . The tube can travel from 13' down the long side of the dock around to a point in line with the short side of the dock. This is 3/4 of a circle of radius 13'. When the rope is lying along the short side of the dock, the remaining possible travel is 1/4 of a circle centered on the other short side corner and this circle has a radius of 13' - 4' = 9' Easy, right? - even though it seemed baffling at first glance.....