document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=238667>Question 238667</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Marginal Cost of Coffee. The manager of a restaurant found that the cost to produce 100 cups of coffee is $11.02, while the cost to produce 400 cups is $40.12. Assume the cost C(x) is a linear function of x, the number of cups produced.\r<BR>\n" );
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document.write( "a. Find a formula for C(x).<BR>\n" );
document.write( "b. What is the fixed cost?<BR>\n" );
document.write( "c.Find the total cost of producing 1000 cups.<BR>\n" );
document.write( "d. Find the total cost of producing 1001 cups.<BR>\n" );
document.write( "e. Find the marginal cost of the 1001st cup.<BR>\n" );
document.write( "f. What is the marginal cost of any cup and what does this mean to the manager?\r<BR>\n" );
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document.write( "I would appreciate any help at all if possible. This is really getting on my nerves. Thanks to who may help me. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #175372 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=175372\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Since it is given that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C(x)\"> is linear, and you are given two points on the Cups(x) vs. Cost(y) graph, use the two-point form of the equation of a line and then put it into slope-intercept form to derive the definition of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C(x)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ \left(\frac{y_1 - y_2}{x_1 - x_2}\right)(x\ -\ x_1) \">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_1,y_1\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_2,y_2\right)\"> are the coordinates of the given points.  For this problem <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(100,11.02\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(400,40.12\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ 11.02\ =\ \left(\frac{11.02 - 40.12}{100 - 400}\right)(x\ -\ 100) \">\r<BR>\n" );
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document.write( "You can do your own arithmetic to verify, but the slope is 0.097 and the y-intercept, which is equal to the fixed cost in this example is 1.32.\r<BR>\n" );
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document.write( "So the slope-intercept, or function of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> form is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 0.097x\ +\ 1.32\">\r<BR>\n" );
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document.write( "Substitute 1000 for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> and calculate, then substitute 1001 for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> and calculate, then calculate the difference.  Given proper arithmetic since this is a linear function, the marginal value at 1001 will be the same as the marginal value at any other value of the independent variable, namely the value of the slope.\r<BR>\n" );
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document.write( "Marginal value is generally defined as the first derivative of the total cost function.  But in the case of a linear function, the first derivative and the slope are the same thing.\r<BR>\n" );
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document.write( "It means that it costs him less than a penny for each cup of coffee he serves after he as recouped the fixed cost of $1.32.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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