document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197606>Question 1197606</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A vendor makes and sells x strings of beads at her souvenir store every<BR>\n" );
document.write( "month. The monthly cost and price  -  demand equations are, respectively,<BR>\n" );
document.write( "C(x) = 0.00002x3 + 0.04x2 + 150x + 45000<BR>\n" );
document.write( "x = 4000  -  200p\r<BR>\n" );
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document.write( " Given the information above<BR>\n" );
document.write( "i. How many sets of beads should she manufacture each month to<BR>\n" );
document.write( "maximize her profit? <BR>\n" );
document.write( "ii. What is the maximum monthly profit? <BR>\n" );
document.write( "iii. How much should the company charge for each set of beads to<BR>\n" );
document.write( "achieve maximum profit? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831072 by </B><A HREF=/tutors/aboutme.mpl?userid=Solver92311><B>Solver92311(821)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Solver92311><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=Solver92311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "First, find the domain of the demand function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \(x(p)\ =\ 4000\,-\,200p\)\">:  If the price is zero, the demand would be 4000, but the revenue would be zero, so zero has to be a lower limit on the demand function (unless you are going to pay customers to take your stuff).  On the other hand, if the price is 20, the demand is zero so you don't sell any.\r<BR>\n" );
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document.write( "You are given the Cost as a function of the number sold, namely:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  C(x(p))\ =\ 0.00002x^3\,+\,0.04x^2\,+\,150x\,+\,45000\">\r<BR>\n" );
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document.write( "The Revenue as a function of the number sold is simple, i.e, the price times the number sold or:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ R(x(p))\ =\ px(p)\">\r<BR>\n" );
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document.write( "And the profit as a function of the number sold is just the Revenue minus the Cost:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(x(p))\ =\ R(x(p))\ -\ C(x(p))\">\r<BR>\n" );
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document.write( "But to know the number of beads to sell to maximize the profit, you need to know the price that maximizes the profit.  Hence, you have to maximize <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(p)\"> and to find that function, substitute the demand function for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(p)\ = \ P(x(p))\ =\ R(4000\,-\,200p)\ -\ C(4000\,-\,200p)\ =\ 4000p\ -\ 200p^2\ -\ \[0.00002(4000\,-\,200p)^3\,+\,0.04(4000\,-\,200p)^2\,+\,150(4000\,-\,200p)\,+\,45000\]\">\r<BR>\n" );
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document.write( "The next step is to simplify the Profit function to a cubic polynomial in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\">.\r<BR>\n" );
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document.write( "Then calculate the first derivative and solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{dP(p)}{dp}\ =\ 0\"> for a value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p\"> in the domain for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> you established at the beginning of the problem.\r<BR>\n" );
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document.write( "And then make sure the second derivative is negative in the interval of the domain of the demand function.\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> <BR>\n" );
document.write( "I > Ø<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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