document.write( "Question 1178968: A metal band is wrapped tightly
\n" ); document.write( "around pipes of radius 3cm and
\n" ); document.write( "9cm. What is the length of the
\n" ); document.write( "band? Express your answer in
\n" ); document.write( "simplest radical form.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #808451 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
Question 1178968
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Extend the two tangent lines until they meet left of the smaller circle.\r\n" );
document.write( "Draw a line from that point to the center of the larger circle.\r\n" );
document.write( "Draw in a radius in each circle to the points of tangency:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We have two similar right triangles.\r\n" );
document.write( "We let h be the length of the horizontal part that extends left of the\r\n" );
document.write( "small circle.\r\n" );
document.write( "\r\n" );
document.write( "By ratios of corresponding sides of similar right triangles,\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "Cross-multiply\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by 3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we know the lengths of the two hypotenuses:\r\n" );
document.write( "\r\n" );
document.write( "The hypotenuse of the small right triangle = h+3 or 3+3=6.\r\n" );
document.write( "The hypotenuse of the large right triangle = h+3+3+9 or h+15 or 3+15 = 18.\r\n" );
document.write( "\r\n" );
document.write( "The upper straight part of the band equals:\r\n" );
document.write( "the longer leg of the large right triangle MINUS\r\n" );
document.write( "the longer leg of the small right triangle.\r\n" );
document.write( "\r\n" );
document.write( "Since the short leg in each triangle is half of the hypotenuse, we\r\n" );
document.write( "know that both right triangles are 30-60-90 right triangles and that\r\n" );
document.write( "means the longer leg is the short leg multiplied by .\r\n" );
document.write( "\r\n" );
document.write( "The long leg of the large right triangle is .\r\n" );
document.write( "The long leg of the small right triangle is .\r\n" );
document.write( "So the upper straight part of the band is  -  = .\r\n" );
document.write( "The lower straight part of the band is also .\r\n" );
document.write( "So both straight parts of the band are . \r\n" );
document.write( "\r\n" );
document.write( "Now we only need to find the curved parts of the band.\r\n" );
document.write( "We find the curved part around the small circle.\r\n" );
document.write( "Since the triangle is a 30-60-90 right triangles, the\r\n" );
document.write( "upper curved part is subtended by a 60-degree angle.\r\n" );
document.write( "So it is 60/360 or 1/6 of the circumference of the\r\n" );
document.write( "entire small circle.  Using C=2πr = 2π(3) = 6π.\r\n" );
document.write( "And 1/6 of that is π.\r\n" );
document.write( "The lower curved part of the band is also π,\r\n" );
document.write( "So the curved part of the band around the small\r\n" );
document.write( "circle is 2π\r\n" );
document.write( "\r\n" );
document.write( "We find the curved part around the large circle.\r\n" );
document.write( "Since the triangle is 30-60-90 right triangles, the\r\n" );
document.write( "upper curved part is subtended by a (180-60) degree angle,\r\n" );
document.write( "or a 120-degree angle.\r\n" );
document.write( "So it is 120/360 or 1/3 of the circumference of the\r\n" );
document.write( "entire large circle.  Using C=2πr = 2π(9) = 18π.\r\n" );
document.write( "And 1/3 of that is 6π.\r\n" );
document.write( "The lower curved part of the band is also 6π,\r\n" );
document.write( "So the curved part of the band around the large\r\n" );
document.write( "circle is 12π\r\n" );
document.write( "\r\n" );
document.write( "Adding the curved parts of the band, we get 2π+12π=14π\r\n" );
document.write( "\r\n" );
document.write( "Adding the curved and straight parts of the band, we get\r\n" );
document.write( "\r\n" );
document.write( "  <-- answer\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );