document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=950160>Question 950160</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Bethany Link delivers parts for several local auto parts stores. She charges clients $.80 per mile driven. She has determined that if she drives 2,350 miles in a month, her average operating cost is $.76 per mile. If Bethany drives 3,350 miles in a month, her average operating cost is $.56 per mile.<BR>\n" );
document.write( "Variable Cost per Unit?<BR>\n" );
document.write( "Fixed Cost?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #580237 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=580237\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> she charges .80 per mile.\r<BR>\n" );
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document.write( "that's her revenue.\r<BR>\n" );
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document.write( "her costs have a fixed component and a per mile component.\r<BR>\n" );
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document.write( "her cost formula would be c = f + v * m\r<BR>\n" );
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document.write( "c/m * m is the total cost.\r<BR>\n" );
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document.write( "it's the cost per mile * the number of miles.\r<BR>\n" );
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document.write( "you'll see why this is shown this way as we go through the specific formulas below.\r<BR>\n" );
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document.write( "f is the fixed cost.\r<BR>\n" );
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document.write( "v is the variable cost per mile.\r<BR>\n" );
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document.write( "m is the number of miles.\r<BR>\n" );
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document.write( "She has determined that if she drives 2,350 miles in a month, her average operating cost is $.76 per mile.\r<BR>\n" );
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document.write( "c/m is equal to .76\r<BR>\n" );
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document.write( "c is equal to c/m * m which is equal to .76 * 2350 = 1786.\r<BR>\n" );
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document.write( "the formula of c = f + v * m becomes:\r<BR>\n" );
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document.write( "1786 = f + 2350 * v\r<BR>\n" );
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document.write( "If Bethany drives 3,350 miles in a month, her average operating cost is $.56 per mile.\r<BR>\n" );
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document.write( "c/m = .56\r<BR>\n" );
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document.write( "c = c/m * m which is equal to .56 * 3350 = 1876\r<BR>\n" );
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document.write( "the formula of c = f + v*m becomes:\r<BR>\n" );
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document.write( "1876 = f + 3350 * v\r<BR>\n" );
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document.write( "she has 2 formulas to work with:\r<BR>\n" );
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document.write( "first formula is:\r<BR>\n" );
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document.write( "1786 = f + 2350 * v\r<BR>\n" );
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document.write( "second formula is:\r<BR>\n" );
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document.write( "1876 = f + 3350 * v\r<BR>\n" );
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document.write( "these 2 formulas need to be solved simultaneously so that she can get a common solution for f and v that applies equally to both of them.\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "1786 = f + 2350 * v (first equation)<BR>\n" );
document.write( "1876 = f + 3350 * v (second equation)\r<BR>\n" );
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document.write( "subtract the first equation from the second equation to get:\r<BR>\n" );
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document.write( "90 = 1000 * v\r<BR>\n" );
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document.write( "solve for v to get:\r<BR>\n" );
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document.write( "v = 90 / 1000 = .09\r<BR>\n" );
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document.write( "replace v in either equation with .09 and solve for f.\r<BR>\n" );
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document.write( "we'll use the first equation, even though the second equation would work just as well.\r<BR>\n" );
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document.write( "you will get the same answer either way as long as you calculated v correctly.\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "1786 = f + 2350 * v (first equation)\r<BR>\n" );
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document.write( "replace v with .09 to get:\r<BR>\n" );
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document.write( "1786 = f + 2350 * .09\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "1786 = f + 211.5\r<BR>\n" );
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document.write( "solve for f to get f = 1786 - 211.5 = 1574.5\r<BR>\n" );
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document.write( "that's the fixed cost.\r<BR>\n" );
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document.write( "the fixed cost is 1574.5<BR>\n" );
document.write( "the variable cost is .09 per mile.\r<BR>\n" );
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document.write( "apply to both equations to see if they are true.\r<BR>\n" );
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document.write( "equations of:\r<BR>\n" );
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document.write( "1786 = f + 2350 * v (first equation)<BR>\n" );
document.write( "1876 = f + 3350 * v (second equation)\r<BR>\n" );
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document.write( "become:\r<BR>\n" );
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document.write( "1786 = 1574.5 + 2350 * .09 (first equation)<BR>\n" );
document.write( "1876 = 1574.5 + 3350 * .09 (second equation)\r<BR>\n" );
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document.write( "evaluate both equations to get:\r<BR>\n" );
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document.write( "1786 = 1786<BR>\n" );
document.write( "1876 = 1876\r<BR>\n" );
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document.write( "this confirms the solutions are correct and apply equally to both equations.\r<BR>\n" );
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document.write( "her fixed cost is 1574.5 and her variable cost is .09 per mile.\r<BR>\n" );
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document.write( "since she charges 80 cents per mile, she makes more profit on the long runs than on the short runs.\r<BR>\n" );
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document.write( "this is because her average cost per mile is lower on the long runs than on the short runs.\r<BR>\n" );
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document.write( "this has to do with the relationship between her fixed costs and her variable costs.\r<BR>\n" );
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document.write( "lower fixed costs and the relative costs would have been closer to each other.\r<BR>\n" );
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document.write( "higher fixed costs and the relative costs would have been further apart from each other.\r<BR>\n" );
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document.write( "the wrinkle in this problem is that they gave you the cost in terms of an average cost per mile.\r<BR>\n" );
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document.write( "you needed to translate that to a total cost by multiplying it by the miles in order for the formula to work.\r<BR>\n" );
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document.write( "working back, when you find the total cost, you need to divide it by the number of miles to get the total cost per mile.\r<BR>\n" );
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document.write( "you can stop here because you already have the solution to your problem.\r<BR>\n" );
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document.write( "that solution is that the fixed cost is 1574.5 and the variable cost is .09 per mile.\r<BR>\n" );
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document.write( "if you want to go further for some additional information, you may, but you don't have to.\r<BR>\n" );
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document.write( "with the information that you have, you can calculate her break even point.\r<BR>\n" );
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document.write( "that would be the point where her cost is equal to her revenue.\r<BR>\n" );
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document.write( "use the equation .80 * x = 1574.5 + .09 * x\r<BR>\n" );
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document.write( "x is the number of miles in this equation.\r<BR>\n" );
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document.write( "solve for x to get x = 2217.605634 miles.\r<BR>\n" );
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document.write( "if you plug that value in the formula, you would get:\r<BR>\n" );
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document.write( ".80 * x = 1574.5 + .09 * x would become:\r<BR>\n" );
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document.write( ".80 * 2217.605634 = 1574.5 + .09 * 2217.605634 which would become:\r<BR>\n" );
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document.write( "1774.084507 = 1774.084507\r<BR>\n" );
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document.write( "this confirms that the break even point is at 2217.605634 miles.\r<BR>\n" );
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document.write( "at that distance she would break even.\r<BR>\n" );
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document.write( "at greater than that distance she makes money.\r<BR>\n" );
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document.write( "at less than that distance she loses money.\r<BR>\n" );
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