document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204444>Question 204444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.  A rectangular field is to be enclosed by 600m of fence.  What dimensions will give the maximum area?  What is the maximum area?\r<BR>\n" );
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document.write( "2.  The captain of a riverboat charges $36 per person including lunch.  The cruise averages 300 customers a day.  The captain is considering increasing the price.  A survey of the customers indicates that for every $2 increase, there would be 10 fewer customers.  What increase in price would maximize the revenue? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154352 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=154352\">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( "The perimeter of a rectangle is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P=2l + 2w\">\r<BR>\n" );
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document.write( "which can be restated as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ l = \frac{P - 2w}{2}\">\r<BR>\n" );
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document.write( "The area of a rectangle is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A = lw\">\r<BR>\n" );
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document.write( "Substituting:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(w) = \left(\frac{P - 2w}{2}\right)w = \frac{Pw - 2w^2}{2}\">\r<BR>\n" );
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document.write( "This function has an extremum where the first derivative is equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dA}{dw} = \frac{P}{2} - 2w\">\r<BR>\n" );
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document.write( "Setting the first derivative equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{P}{2} - 2w = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{P}{2} = 2w\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w = \frac{P}{4}\">\r<BR>\n" );
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document.write( "This extremum is a maximum if the value of the 2nd derivative is negative.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2A}{dw^2} = -2\">\r<BR>\n" );
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document.write( "Which is negative for all values of the independent variable, hence the extremum is a maximum.\r<BR>\n" );
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document.write( "In order for the width to be one fourth of the perimeter, the length must also be one-fourth of the perimeter.  That is to say, the maximum area for a rectangle with a given perimeter is achieved when the rectangle is a square with side measure of one-fourth of the perimeter.\r<BR>\n" );
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document.write( "=====================================================\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> represent the price of one fare.  Then the revenue function is the number of passengers times the price of a fare.  The price of one fare minus 36, quantity divided by 2 is the number of $2 increases to the price. So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ R(p) = \left(300 - 10\left(\frac{p-36}{2}\right)\right)p\">\r<BR>\n" );
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document.write( "Which simplifies to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 480p - 5p^2\">\r<BR>\n" );
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document.write( "First derivative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dR}{dp}=480 - 10p\">\r<BR>\n" );
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document.write( "Set equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 480 = 10p \ \ \Rightarrow\ \ p = 48\">\r<BR>\n" );
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document.write( "Second derivative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2R}{dp^2}=-10\">\r<BR>\n" );
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document.write( "Hence extreme point is a maximum.\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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