document.write( "Question 1206903: A small radio transmitter broadcasts in a 21 mile radius. If you drive along a straight line from a city 25 miles north of the transmitter to a second city 29 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter? \n" ); document.write( "

Algebra.Com's Answer #844656 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A small radio transmitter broadcasts in a 21 mile radius. If you drive along a straight line
\n" ); document.write( "from a city 25 miles north of the transmitter to a second city 29 miles east of the transmitter,
\n" ); document.write( "during how much of the drive will you pick up a signal from the transmitter?
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\n" ); document.write( "\n" ); document.write( " This problem admits simple and elegant geometric solution without using equations.\r
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document.write( "Let A be the point 25 miles north from the transmitter.\r\n" );
document.write( "Let B be the point 29 miles east  from the transmitter.\r\n" );
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document.write( "Let O be the circle of the radius 21 miles with the center at the transmitter location.\r\n" );
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document.write( "Then we have a right angled triangle OAB and straight line AB, intersecting this triangle \r\n" );
document.write( "in points B and C.  They want we find the length of the chord BC.\r\n" );
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document.write( "My designations are close to that on the plot of the other tutor.\r\n" );
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document.write( "Draw the perpendicular OE from the center/vertex O to line BC.\r\n" );
document.write( "This perpendicular is the height (the altitude) in triangle OAB on the hypotenuse.\r\n" );
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document.write( "The area of this triangle can be computed as half the product of its legs\r\n" );
document.write( "or as half the product of the hypotenuse and the height.  So, we can write this equation\r\n" );
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document.write( "    |AB|*h = |OA|*|OB|,     (1)\r\n" );
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document.write( "where h is the height OE.  The hypotenuse |AB| is   = 38.2884 miles  (rounded).  From (1)\r\n" );
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document.write( "    h = .\r\n" );
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document.write( "It gives the distance from the center O to line AB.\r\n" );
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document.write( "Then the chord length is\r\n" );
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document.write( "    |BC| = 2*|BE| = 2*|CE| =  =  = 18.1611  miles (rounded).\r\n" );
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document.write( "Now we have all the numbers and the ANSWER:  the entire trip is about 38.2884 miles;\r\n" );
document.write( "                                             the part where the signal is accessible is about 18.1611  miles;\r\n" );
document.write( "                                             the ratio of these values is   = 0.4743 (rounded).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "At the beginning, I said that the equations are not used in the solution, but it is not precisely correct.
\n" ); document.write( "One equation (1) is still used, but in minimal degree.
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