document.write( "Question 1180635: Two pulleys are connected by a belt. The radii of the pulleys are 6 cm and 21 cm, and the distance between
\n" ); document.write( "their centers is 30 cm. Find the total length of belt needed to connect the pulleys. \n" ); document.write( "

Algebra.Com's Answer #810429 by ikleyn(52810) $\"\"$ $\"About$

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\n" ); document.write( "Two pulleys are connected by a belt. The radii of the pulleys are 6 cm and 21 cm, and the distance between
\n" ); document.write( "their centers is 30 cm. Find the total length of belt needed to connect the pulleys.
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\n" ); document.write( "\n" ); document.write( "            @MathLover used the formula from this web-site\r
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\n" ); document.write( "\n" ); document.write( "            https://www.sudenga.com/practical-applications/figuring-belt-lengths-and-distance-between-pulleys \r
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\n" ); document.write( "\n" ); document.write( "            without referencing.\r
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\n" ); document.write( "\n" ); document.write( "            At that web-site, the formula is given without derivation or explanations.\r
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\n" ); document.write( "\n" ); document.write( "            Therefore, I developed here the full solution with detailed derivation of all formulas -
\n" ); document.write( "            - partly to check her calculations and partly to have a full derivation.\r
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\n" ); document.write( "\n" ); document.write( "            Our results are close enough.\r
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document.write( "The key here is to note that the point at which the belt just leaves or enters the pulleys, is a point of tangency, \r\n" );
document.write( "that is, the segment connecting these points on the two pulleys are at right angles to the radii of both pulleys. \r\n" );
document.write( "Let this segment length be  x cm for my following calculations.  Draw a line starting from the center of the  6 cm pulley parallel \r\n" );
document.write( "to the tangent line until it intersects with the  21 cm radius of the larger pulley. \r\n" );
document.write( "You now have a right triangle with legs  x  and  15 cm = 21 cm - 6 cm, with hypotenuse the given distance between the centers \r\n" );
document.write( "of the pulleys = 30 cm. \r\n" );
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document.write( "You now can calculate with the Pythagorean theorem   x =  =  = 25.98 cm.\r\n" );
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document.write( "All you need to do next is calculate the rest of the belt that remains on the pulleys.\r\n" );
document.write( "By using the (x,15,30) right triangle, you can determine the central angles of the arcs of each pulley that the belts are on. \r\n" );
document.write( "Note for the small pulley the central angle is less than 180°, and greater than 180° for the larger pulley. \r\n" );
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document.write( "The acute angle of this triangle at the center of larger pulley is  \r\n" );
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document.write( "        a =  =  =  =  = 1.0472 radians, or 60°.\r\n" );
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document.write( "That part of the belt remaining on the pulleys can now be calculated by multiplying the central angle by  \r\n" );
document.write( "the radius of the respective pulleys. \r\n" );
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document.write( "For the greater pulley, the central angle is   radians; \r\n" );
document.write( "                        THEREFORE, the belt remaining on the greater pulley is of  G = 2*(3.14159-1.0472)*21 = 87.9644 cm long.\r\n" );
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document.write( "For the greater pulley, the central angle is   radians; \r\n" );
document.write( "                        THEREFORE, the belt remaining on the greater pulley is of  S = 2*1.0472*6 = 12.56 cm long.\r\n" );
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document.write( "Now the total length of belt is the sum  2x + G + S =  2*25.98 + 87.96 + 12.56 = 157.04 cm.\r\n" );
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