document.write( "Question 392634: a surveyor wishes to find the distance between two inaccessible points A&B on opposite side of the lake while standing at point C she finds that AC=259m ,BC=423inches and angle ABC measures 132 40' find the distance AB? \n" ); document.write( "
Algebra.Com's Answer #279399 by jsmallt9(3758)\"\" \"About 
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The Law of Cosines has four variables: the lengths of the 3 sides of a triangle and the measure of one angle.

\n" ); document.write( "Your problem gives you 2 sides and the measure of one angle. So you know the values 3 of the 4 variables in the Law of Cosines. Anytime you know all but 1 variable in an equation you should be able to solve for the value of that last unknown. So we will use the Law of Cosines to solve this problem:
\n" ); document.write( "\"%28AC%29%5E2+=+%28AB%29%5E2+%2B+%28BC%29%5E2+-+2%28AB%29%28BC%29cos%28B%29\"
\n" ); document.write( "To convert BC into meters, we divide by 39.37:
\n" ); document.write( "BC = 423 inches = 432/39.37 meters = 10.7442214884429769 meters.
\n" ); document.write( "Also 132 degrees and 40 minutes = 132.67 degrees since 40 minutes is 2/3 of a degree and 2/3 as a decimal is approximately 0.67.
\n" ); document.write( "Inserting these values into our Law of Cosines equation we get:
\n" ); document.write( "
\n" ); document.write( "To avoid confusion I am going to replace (AB) with an \"x\":
\n" ); document.write( "
\n" ); document.write( "Simplifying we get:
\n" ); document.write( "
\n" ); document.write( "\"67081+=+x%5E2+%2B+115.4382953927198178+-+%28-14.5643246369116504x%29\"
\n" ); document.write( "\"67081+=+x%5E2+%2B+115.4382953927198178+%2B+14.5643246369116504x\"
\n" ); document.write( "This is a quadratic equation because of the \"x%5E2\". SO we want one side to be zero. Subtracting 67081 from each side we get:
\n" ); document.write( "\"0+=+x%5E2+%2B+14.8742038845055153x+-66975.5617046072801800\"
\n" ); document.write( "Now we can use the Quadratic Formula:
\n" ); document.write( "
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\"x+=+%28-%2814.8742038845055153%29+%2B-+sqrt%28268123.4887596269597000%29%29%2F2%281%29\"
\n" ); document.write( "\"x+=+%28-14.8742038845055153+%2B-+sqrt%28268123.4887596269597000%29%29%2F2\"
\n" ); document.write( "In long form this is:
\n" ); document.write( "\"x+=+%28-14.8742038845055153+%2B+sqrt%28268123.4887596269597000%29%29%2F2\" or \"x+=+%28-14.8742038845055153+-+sqrt%28268123.4887596269597000%29%29%2F2\"
\n" ); document.write( "We can see that the second equation will give us a negative value. But the side of a triangle cannot be negative. So we will reject the second solution. The only solution we can use is:
\n" ); document.write( "\"x+=+%28-14.8742038845055153+%2B+sqrt%28268123.4887596269597000%29%29%2F2\"
\n" ); document.write( "Using a calculator to find the square root we get:
\n" ); document.write( "\"x+=+%28-14.8742038845055153+%2B+517.8064201606880036%29%2F2\"
\n" ); document.write( "\"x+=+502.9322162761824883%2F2\"
\n" ); document.write( "x = 251.4661081380912442
\n" ); document.write( "So AB is approximately 251.5 meters long.
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