document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199350>Question 1199350</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A high-technology electronics firm needs a special microprocessor for use in a microcomputer it manufactures. Three alternatives<BR>\n" );
document.write( "have been identified for satisfying its needs. It can purchase the microprocessor from a supplier at a cost of $10 each. The firm can also<BR>\n" );
document.write( "purchase one of two pieces of automated equipment and manufacture the microprocessors. One piece of equipment costs $80,000 and would<BR>\n" );
document.write( "have variable costs per microprocessor of $8. A more highly automated piece of equipment costs $120, 000 and would result in variable cost of<BR>\n" );
document.write( "$5/unit. Determine the minimum cost alternative for different ranges of output.<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 #833208 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=833208\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the 3 alternatives are:<BR>\n" );
document.write( "y =  10 * x<BR>\n" );
document.write( "y = 80000 + 8 * x<BR>\n" );
document.write( "y = 120000 + 120000 + 5 * x<BR>\n" );
document.write( "y is the total costs.<BR>\n" );
document.write( "x is the number of units.\r<BR>\n" );
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document.write( "if you have a graphing calculator, such as he one at <a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a>, you can get a visual idea of what the answer will be, as shown below:\r<BR>\n" );
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document.write( "you can see that y = 10x is the minimum cost solution from 0 < x < 24000 and y = 120000 + 5x from 24000 < x < infinity.\r<BR>\n" );
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document.write( "to find the answer algebraically, i think the best way is to find the breaking points and then look at the intervals in between.\r<BR>\n" );
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document.write( "first compare y = 10x to y = 80,000 + 8x to see at what point they break even.<BR>\n" );
document.write( "this happens when 10x = 80,000 + 8x.<BR>\n" );
document.write( "subtract 8x from both sides of the equation to get:<BR>\n" );
document.write( "2x = 80,000.<BR>\n" );
document.write( "solve for x to get x = 40,000.\r<BR>\n" );
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document.write( "next compare y = 10x to y = 120,000 + 5x to see at what point they break even.<BR>\n" );
document.write( "this happens when 10x = 120,000 + 5x.<BR>\n" );
document.write( "subtract 5x from both sides of the equation to get:<BR>\n" );
document.write( "5x = 120,000<BR>\n" );
document.write( "solve for x to get x = 24,000.\r<BR>\n" );
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document.write( "next compare y = 80,000 + 8x to y = 120,000 + 5x to see at what point they break even.<BR>\n" );
document.write( "this happens when 80,000 + 8x = 120,000 + 5x<BR>\n" );
document.write( "subtract 5x from both sides of the equation and subtract 80,000 from both sides of the equation to get:<BR>\n" );
document.write( "3x = 40,000<BR>\n" );
document.write( "solve for x to get x = 13,333 and 1/3.\r<BR>\n" );
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document.write( "you have 3 breaking points.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "x = 40,000<BR>\n" );
document.write( "x = 24,000<BR>\n" );
document.write( "x = 13,333 and 1/3\r<BR>\n" );
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document.write( "your intervals will be:<BR>\n" );
document.write( "0 < x < 13,333 and 1/3<BR>\n" );
document.write( "13,333 and 1/3 < x < 24,000<BR>\n" );
document.write( "24,000 < x < 40,000<BR>\n" );
document.write( "40,000 < x < infinity.\r<BR>\n" );
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document.write( "you can pick any value within these intervals to see which is the minimum cost.<BR>\n" );
document.write( "i usually pick a value that's easy to calculate within each interval.\r<BR>\n" );
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document.write( "at x = 10,000, which is in the interval from 0 to 13,333 and 1/3, you get:<BR>\n" );
document.write( "y = 10x = 100,000<BR>\n" );
document.write( "y = 80,000 + 8x = 160,000<BR>\n" );
document.write( "y = 120,000 + 5x = 170,000<BR>\n" );
document.write( "y = 10x is the minimum cost equation in this interval.\r<BR>\n" );
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document.write( "at x = 20,000, which is in the interval from 13,333 and 1/3 < x < 24,000, you get:\r<BR>\n" );
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document.write( "y = 10x = 200,000<BR>\n" );
document.write( "y = 80,000 + 8x = 240,000<BR>\n" );
document.write( "y = 120,000 + 5x = 220,000<BR>\n" );
document.write( "y = 10x is the minimum cost equation in this interval.\r<BR>\n" );
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document.write( "at x = 30,000, which is in the interval from 24,000 < x < 40,000, you get:\r<BR>\n" );
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document.write( "y = 10x = 300,000<BR>\n" );
document.write( "y = 80,000 + 8x = 320,000<BR>\n" );
document.write( "y = 120,000 + 5x = 270,000<BR>\n" );
document.write( "y = 120,000 + 5x is the minimum cost equation in this interval.\r<BR>\n" );
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document.write( "at x = 50,000, which is in the interval from 40,000 < x < infinity, you get:<BR>\n" );
document.write( "y = 10x = 500,000<BR>\n" );
document.write( "y = 80,000 + 8x = 480,000<BR>\n" );
document.write( "y = 120,000 + 5x = 370,000<BR>\n" );
document.write( "y = 120,000 + 5x is the minimum cost eqution in this interval.\r<BR>\n" );
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document.write( "what you get is:\r<BR>\n" );
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document.write( "y = 10x is the minimum cost equation from 0 < x < 24,000<BR>\n" );
document.write( "y = 120,000 + 5x is the minimum cost equation from 24,000 < x < infinity.\r<BR>\n" );
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document.write( "the algebraic solution agrees with the graphical solution, as you can determine by looking at the graph and looking at the graphical solution within each interval.\r<BR>\n" );
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document.write( "in graph, .....<BR>\n" );
document.write( "y = 10x is red<BR>\n" );
document.write( "y = 80,000 + 8x is blue<BR>\n" );
document.write( "y = 120,000 + 5x is green.<BR>\n" );
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