document.write( "Question 225993: A boat is pulled toward dock by means of a rope wound on a drum that is located 6ft above the bow of the boat. if the rope is being pulled in at the rate of 6ft/sec, how fast is the boat approaching the dock when it is 24ft from the dock? \n" ); document.write( "
Algebra.Com's Answer #168459 by Edwin McCravy(20056)\"\" \"About 
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A boat is pulled toward dock by means of a rope wound on a drum that is located 6ft above the bow of the boat. if the rope is being pulled in at the rate of 6ft/sec, how fast is the boat approaching the dock when it is 24ft from the dock?
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document.write( "We can describe the triangle by the Pythagorean theorem:\r\n" );
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document.write( "               \"x%5E2%2B6%5E2=r%5E2\"\r\n" );
document.write( "                \"x%5E2%2B36=r%5E2\"\r\n" );
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document.write( "We take the derivative implicitly:\r\n" );
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document.write( "               \"2x%28dx%2Fdt%29+%2B+0+=+2r%28dr%2Fdt%29\" \r\n" );
document.write( "                   \"2x%28dx%2Fdt%29+=+2r%28dr%2Fdt%29\"\r\n" );
document.write( "                    \"x%28dx%2Fdt%29+=+r%28dr%2Fdt%29\"\r\n" );
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document.write( "We are told that \"r\" (the rope length) is shrinking at the \r\n" );
document.write( "constant rate of 6 ft/sec.  Therefore \"%28dr%29%2F%28dt%29=-6\"\r\n" );
document.write( "(it is negative because \"r\" is shrinking).  So we\r\n" );
document.write( "substitute \"-6\" for \"dr%2Fdt\"\r\n" );
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document.write( "                    \"x%28dx%2Fdt%29+=+r%28-6%29\"\r\n" );
document.write( "                    \"x%28dx%2Fdt%29+=+-6r\"\r\n" );
document.write( "                      \"dx%2Fdt+=+-6r%2Fx%29\"\r\n" );
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document.write( "Now we want to freeze the motion at the instant when the boat\r\n" );
document.write( "is 24 feet from the dock.  That is, when x = 24.\r\n" );
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document.write( "Since               \"x%5E2%2B36=r%5E2\"\r\n" );
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document.write( "we substitute \"x=24\"\r\n" );
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document.write( "                \"24%5E2%2B36=r%5E2\"\r\n" );
document.write( "                \"576%2B36=r%5E2\"\r\n" );
document.write( "                   \"612=r%5E2\"\r\n" );
document.write( "                  \"sqrt%28612%29=r\"\r\n" );
document.write( "                \"sqrt%2836%2A17%29=r\"   \r\n" );
document.write( "                  \"6sqrt%2817%29=r\"\r\n" );
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document.write( "So we substitute \"6sqrt%2817%29\" for \"r\" and\r\n" );
document.write( "\"24\" for \"x\" in\r\n" );
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document.write( "               \"dx%2Fdt+=+-6r%2Fx%29\"\r\n" );
document.write( "               \"dx%2Fdt+=+-6%286sqrt%2817%29%2F24%29\"\r\n" );
document.write( "               \"dx%2Fdt+=+-%2836sqrt%2817%29%2F24%29\"\r\n" );
document.write( "               \"dx%2Fdt+=+-%2836sqrt%2817%29%2F24%29\"\r\n" );
document.write( "               \"dx%2Fdt+=+-%283sqrt%2817%29%2F2%29\"\r\n" );
document.write( "               \"dx%2Fdt+=+-6.2\", approximately.\r\n" );
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document.write( "So x is shrinking at the rate of 6.2 ft/sec. Therefore\r\n" );
document.write( "that is how fast the boat is approaching the dock at\r\n" );
document.write( "that instant.              \r\n" );
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document.write( "Edwin
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