document.write( "Question 1185007: A chord of a circle of diameter 10 feet is decreasing in length 1 foot per minute. Find the rate of change of the smaller arc subtended by the chord when the chord is 8 feet long. \n" ); document.write( "

Algebra.Com's Answer #815962 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "The diameter is 10 ft., so the radius is 5 ft.\r\n" );
document.write( "Let the arc at the top be \"s\".  We will need the formula \r\n" );
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document.write( "and its derivative with respect to time t:\r\n" );
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document.write( "Let the length of the chord be x. Since it is decreasing in length 1 foot per\r\n" );
document.write( "minute, taken negative since it is decreasing.\r\n" );
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document.write( "Next we have to find an equation for x in terms of the chord and θ.  So we draw\r\n" );
document.write( "this green line perpendicular to the chord, which bisects the chord and θ\r\n" );
document.write( "and gives us two congruent right triangles.\r\n" );
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document.write( "From the right triangle,\r\n" );
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document.write( "Take the derivative with respect to time t:\r\n" );
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document.write( "Since \r\n" );
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document.write( "We redraw the figure at the instant when the chord x = 8, which\r\n" );
document.write( "makes the left half of it 4.  And since it's a 3-4-5 right triangle,\r\n" );
document.write( "the green line is 3.\r\n" );
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document.write( "Since \r\n" );
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document.write( "Finally we substitute in\r\n" );
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document.write( "The negative sign indicates that the arc is decreasing\r\n" );
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document.write( "Edwin

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