A thin belt is stretched around three pulleys, each of which is 2 feet in diameter.
The distances between the centers of the pulleys are 6 feet, 9 feet, and 13 feet.
How long is the belt?
The portions of the belt that are not touching the pulleys go from point of tangency to point of tangency, therefore making right angles with radii of the pulleys. With those radii they make rectangles with the vertices being the points of tangency and the centers of the pulleys, and so the total of these portions is 6+9+13=28 feet, as the sides connecting the points of tangency equal the sides connecting the pulley centers.
The portion in contact with a given pulley has an arc whose central angle is the supplement of the angle of the triangle that is at the center of that pulley (the above-mentioned rectangles occupy 180° of the pulley's circle and the angle of the triangle together with the arc of contact together make up the other 180°). The total angles are therefore 180-A + 180-B + 180-C = 3*180 - 180 = 360°, where A, B and C are the measures of the angles of the triangle. That makes these portions of the belt length total to the full circumference of a circle of radius 1 foot, and therefore equal to 2*pi.
The total belt length is therefore 28 + 2*pi feet.
Edited on May 9, 2017, 9:39 am
Posted by Charlie
on 2017-05-09 09:35:50