document.write( "Question 214882: The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley. The point on the abandoned road closest to Foxton is 40 east of Grimley.
\n" ); document.write( "County officials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost $100,000 per mile, whereas building a new road would cost $200,000 per mile. How much of the abandoned road should be used if the officials intend to spend $6.8 million? Would it cost less than this amount to build a new road connecting the towns directly? \n" ); document.write( "
Algebra.Com's Answer #162369 by ankor@dixie-net.com(22740)\"\" \"About 
You can put this solution on YOUR website!
The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley.
\n" ); document.write( "The point on the abandoned road closest to Foxton is 40 east of Grimley.
\n" ); document.write( "County officials are about to build a new road connecting the two towns.
\n" ); document.write( " They have determined that restoring the old road would cost $100,000 per mile,
\n" ); document.write( " whereas building a new road would cost $200,000 per mile.
\n" ); document.write( "How much of the abandoned road should be used if the officials intend to spend $6.8 million?
\n" ); document.write( ";
\n" ); document.write( "Drawing a diagram of this will make it easy to understand.
\n" ); document.write( ":
\n" ); document.write( "Let x = the unused portion of the old road from a point south of Foxton
\n" ); document.write( "then
\n" ); document.write( "(40-x) = the portion of the old road to be used.
\n" ); document.write( "That's the point where it begins the new road diagonally to Foxton
\n" ); document.write( ":
\n" ); document.write( "This diagonal is the hypotenuse of a right triangle formed by sides 10 and x
\n" ); document.write( ":
\n" ); document.write( "The equation using 1000's of dollars
\n" ); document.write( ":
\n" ); document.write( "old road cost + new road cost = 6.8 million
\n" ); document.write( "100(40-x) + 200(\"sqrt%28x%5E2+%2B+10%5E2%29\") = 6800
\n" ); document.write( ":
\n" ); document.write( "4000 - 100x + 200(\"sqrt%28x%5E2+%2B+100%29\") = 6800
\n" ); document.write( ":
\n" ); document.write( "Get the square root by itself on the left
\n" ); document.write( "200(\"sqrt%28x%5E2+%2B+100%29\") = 100x + 6800 - 4000
\n" ); document.write( ":
\n" ); document.write( "200(\"sqrt%28x%5E2+%2B+100%29\") = 100x + 2800
\n" ); document.write( "square both sides
\n" ); document.write( "40000(x^2 + 100) = (100x + 2800)^2
\n" ); document.write( ":
\n" ); document.write( "40000x^2 + 4000000 = 10000x^2 + 560000x + 7840000
\n" ); document.write( ":
\n" ); document.write( "0 = 10000x^2 - 40000x^2 + 560000x + 7840000 - 4000000
\n" ); document.write( ":
\n" ); document.write( "-30000x^2 + 560000x + 3840000 = 0
\n" ); document.write( "Simplify divide by -10000 (change the signs)
\n" ); document.write( "3x^2 - 56x - 384 = 0
\n" ); document.write( "Factor this to:
\n" ); document.write( "(3x + 16)(x - 24) =
\n" ); document.write( "Positive solution
\n" ); document.write( "x = 24 mi of unused old road
\n" ); document.write( "then
\n" ); document.write( "40 - 24 = 16 mi of the old road used at 100k/mi
\n" ); document.write( ":
\n" ); document.write( "New road dist, find the hypotenuse:
\n" ); document.write( "NR = \"sqrt%2824%5E2%2B+10%5E2%29\"
\n" ); document.write( "NR = 26 mi of new road at 200k/mi
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\n" ); document.write( ":
\n" ); document.write( "Check the cost, new road + old road:
\n" ); document.write( "Cost = 200(26) + 100(16)
\n" ); document.write( "Cost = 5200 + 1600
\n" ); document.write( "Cost = 6800 thousand which is 6.8 million
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( " Would it cost less than this amount to build a new road connecting the towns
\n" ); document.write( ":
\n" ); document.write( "No
\n" ); document.write( "cost = 200*\"sqrt%2840%5E2+%2B+10%5E2%29\"
\n" ); document.write( "Cost = 200 * 41.231 mi
\n" ); document.write( "Cost = 8246 thousand which is 8.246 million
\n" ); document.write( ";
\n" ); document.write( ";
\n" ); document.write( "Actually you could find the minimum cost by finding the vertex of the equation, but they are determined to spend 6.8 million, how like the government!! \n" ); document.write( "
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