document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=90413>Question 90413</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0073016020/algebrahomeworkh>Beginning Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> If someone could help me with this word problem I would very appreciative.  \r<BR>\n" );
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document.write( "Business and finance.  In planning for a new item, a manufacturer assumes that the number of items produced x and the cost in dollars C of producing these items are related by a linear equation.  Projections are that 100 items will cost $10,000 to produce and that 300 items will cost $22,000 to produce.  Find the equation that relates C and x.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65670 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=65670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's suppose that we label the y-axis in dollars and the x-axis in items produced.<BR>\n" );
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document.write( "Then our coordinate points will be in the form (x, y) corresponding to (items, dollars).<BR>\n" );
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document.write( "In this form, the problem gives you two points: (100, $10,000) and (300, $22,000).<BR>\n" );
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document.write( "Remember that each of these points is in the form (x,y).<BR>\n" );
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document.write( "Now suppose that we use the slope-intercept form of an equation.  The slope intercept<BR>\n" );
document.write( "form is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+mx+%2B+b ALIGN=MIDDLE  ALT=\"y+=+mx+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "in which m is the slope of the graph and b is the value where the graph crosses the y-axis.<BR>\n" );
document.write( "This is a linear equation, so it will lead us to a linear relationship between cost (C)<BR>\n" );
document.write( "and the number of units produced (x).<BR>\n" );
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document.write( "Let's try to find m and b. Then we can substitute those values into the slope-intercept<BR>\n" );
document.write( "form and we will have the equation.<BR>\n" );
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document.write( "The slope is defined as the amount of change in the y direction between the two points divided<BR>\n" );
document.write( "by the change in the x direction between the two points.  Notice that the change in going<BR>\n" );
document.write( "from the y value of the first point to the y value of the second point is an increase<BR>\n" );
document.write( "from $10,000 to $22,000 ... an increase of $12,000.<BR>\n" );
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document.write( "Similarly, the corresponding change in going from the x value of the first point to the<BR>\n" );
document.write( "x value of the second point is an increase from 100 items to 300 items ... an increase<BR>\n" );
document.write( "of 200 items.<BR>\n" );
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document.write( "Therefore, we can say that the slope is the change in y (a change of $12,000) divided by<BR>\n" );
document.write( "the change in x (a change of 200 items). Therefore, the slope is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+m+=+12000%2F200+=+60 ALIGN=MIDDLE  ALT=\"slope+=+m+=+12000%2F200+=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now that we know the slope (m) is 60, we can go back to the slope-intercept form of the<BR>\n" );
document.write( "equation and substitute 60 for m to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+60%2Ax+%2B+b ALIGN=MIDDLE  ALT=\"y+=+60%2Ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "At this point, let's recognize that the y-value is the cost of production so we can substitute<BR>\n" );
document.write( "C for y. This makes the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+60%2Ax+%2B+b ALIGN=MIDDLE  ALT=\"C+=+60%2Ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "furthermore x represents the number of items produced.<BR>\n" );
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document.write( "All we now need is to find b. We can do that by taking either of the two points and<BR>\n" );
document.write( "substituting the x and C values for that point into the equation we now have.  Let's choose<BR>\n" );
document.write( "the point (100, $10,000). [We could just as well use (300, $22,000)]. Substituting<BR>\n" );
document.write( "100 for x and 10000 for C, our equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000+=+60%2A100+%2B+b ALIGN=MIDDLE  ALT=\"10000+=+60%2A100+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply the 60 times 100 and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000+=+6000+%2B+b ALIGN=MIDDLE  ALT=\"10000+=+6000+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally solve for b by subtracting 6000 from both sides of the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000+-+6000+=+b ALIGN=MIDDLE  ALT=\"10000+-+6000+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+4000 ALIGN=MIDDLE  ALT=\"b+=+4000\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substituting this value of b into our cost equation<BR>\n" );
document.write( "results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+60%2Ax+%2B+4000 ALIGN=MIDDLE  ALT=\"C+=+60%2Ax+%2B+4000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This should be the equation that you are looking for.  Let's check it by taking one of<BR>\n" );
document.write( "the coordinates of the second point, plugging it into the equation, and seeing if it<BR>\n" );
document.write( "gives the other coordinate value of that point.  Recall that we used the point (100, 10000)<BR>\n" );
document.write( "to find the value of b. So we can use the second point (300, 22000) in our check.<BR>\n" );
document.write( "Let's take the x value of 300 and plug that value into our equation.  When we do that the<BR>\n" );
document.write( "Cost equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+60%2A300+%2B+4000 ALIGN=MIDDLE  ALT=\"C+=+60%2A300+%2B+4000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiplying on the right side [ 60 times 300] results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+18000+%2B+4000 ALIGN=MIDDLE  ALT=\"C+=+18000+%2B+4000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally adding the terms on the right side results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+22000 ALIGN=MIDDLE  ALT=\"C+=+22000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That's exactly the answer we should get ... for 300 units the cost is $22,000. So the<BR>\n" );
document.write( "equation checks out.  The equation you are looking for is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+60%2Ax+%2B+4000 ALIGN=MIDDLE  ALT=\"C+=+60%2Ax+%2B+4000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to understand the problem.<BR>\n" );
document.write( ". </SPAN>\n" );
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