document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=473532>Question 473532</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Aki's Bicycle Designs has determined that when x bicycles are built, the average cost per bicycle is given by C(x) =-0.2x^2-0.1x+9.743, where C(x) is in dollars. How many bicycles should the shop build to minimize the average cost per bicycle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #324768 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=324768\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> c(x) = -.2x^2-.1x+9.743<BR>\n" );
document.write( "graph of this equation looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-20%2C20%2C%28-.2%29%2Ax%5E2-%28.1%29%2Ax%2B9.743%2C9.755%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C10%2C-20%2C20%2C%28-.2%29%2Ax%5E2-%28.1%29%2Ax%2B9.743%2C9.755%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "since x can't be negative, then only the right side of the graph is valid (x >= 0).<BR>\n" );
document.write( "it appears that this graph indicates a maximum point and not a minimum point.<BR>\n" );
document.write( "that's because the coefficient of the x^2 term is negative.<BR>\n" );
document.write( "you can see from this graph that the cost per bicycle starts off at a maximum point when x = 0 and then tapers down to 0 when x = somewhere between 6 and 7.<BR>\n" );
document.write( "there is no minimum point.<BR>\n" );
document.write( "only a maximum point.<BR>\n" );
document.write( "the maximum point would be found from the equation:<BR>\n" );
document.write( "x = -b/2a<BR>\n" );
document.write( "in this equation, a = -.2 and b = -.1<BR>\n" );
document.write( "x = -b/2a = -(-.1)/(2*(-.2) which becomes .1/-.4 which becomes -1/4.<BR>\n" );
document.write( "the maximum point of this graph is when x = -.25<BR>\n" );
document.write( "at that point, y would be equal to -.2*(-.25)^2-.1*(-.25)+9.743.<BR>\n" );
document.write( "solving this equation gets you:<BR>\n" );
document.write( "y = 9.755.<BR>\n" );
document.write( "the horizontal line in the graph is at y = 9.755.<BR>\n" );
document.write( "note that this is a maximum point, not a minimum point.<BR>\n" );
document.write( "it looks like the problem is flawed since there is no minimum point.<BR>\n" );
document.write( "it appears the more bicycles that are built, the cheaper the cost, until you get to a point where the cost per bicycle is 0.<BR>\n" );
document.write( "that would be at the upper root of the quadratic equation.<BR>\n" );
document.write( "you would use the quadratic formula to find the exact value of this root.<BR>\n" );
document.write( "the roots of this equation are:<BR>\n" );
document.write( "x = -7.23408906<BR>\n" );
document.write( "x = 6.73408906<BR>\n" );
document.write( "the higher root is equal to 6.73408906.<BR>\n" );
document.write( "the graph shows the approximate location.<BR>\n" );
document.write( "bottom line is you don't really have a minimum cost point with the equation you showed, only a maximum point.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );