document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=373981>Question 373981</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The demand equation for a certain product is given by\r<BR>\n" );
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document.write( "p=5000(1-(4/(4+e^-0.002x)))\r<BR>\n" );
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document.write( "Find the demands x for prices of <BR>\n" );
document.write( "a) p=$600<BR>\n" );
document.write( "b) p=$400 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #266278 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3759)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=266278\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=5000%281-%284%2F%284%2Be%5E%28-0.002x%29%29%29%29 ALIGN=MIDDLE  ALT=\"p=5000%281-%284%2F%284%2Be%5E%28-0.002x%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find x for p = 600:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=600+=+5000%281-%284%2F%284%2Be%5E%28-0.002x%29%29%29%29 ALIGN=MIDDLE  ALT=\"600+=+5000%281-%284%2F%284%2Be%5E%28-0.002x%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We need \"peel away\" from the outside on the right side. First we get rid of the 5000 by dividing both sides be 5000:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=600%2F5000+=+1-%284%2F%284%2Be%5E%28-0.002x%29%29%29 ALIGN=MIDDLE  ALT=\"600%2F5000+=+1-%284%2F%284%2Be%5E%28-0.002x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.12+=+1-%284%2F%284%2Be%5E%28-0.002x%29%29%29 ALIGN=MIDDLE  ALT=\"0.12+=+1-%284%2F%284%2Be%5E%28-0.002x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next the 1 must go. Subtract 1 from each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.88+=+-%284%2F%284%2Be%5E%28-0.002x%29%29%29 ALIGN=MIDDLE  ALT=\"-0.88+=+-%284%2F%284%2Be%5E%28-0.002x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply (or divide) both sides by -1 to eliminate the minus sign:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.88+=+4%2F%284%2Be%5E%28-0.002x%29%29 ALIGN=MIDDLE  ALT=\"0.88+=+4%2F%284%2Be%5E%28-0.002x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we'll eliminate the fraction by multiplying both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Be%5E%28-0.002x%29%29 ALIGN=MIDDLE  ALT=\"%284%2Be%5E%28-0.002x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.88%29%2A%284%2Be%5E%28-0.002x%29%29+=+4 ALIGN=MIDDLE  ALT=\"0.88%29%2A%284%2Be%5E%28-0.002x%29%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Distributive Property on the left side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.88%29%2A%284%29%2B%280.88%29%2Ae%5E%28-0.002x%29+=+4 ALIGN=MIDDLE  ALT=\"0.88%29%2A%284%29%2B%280.88%29%2Ae%5E%28-0.002x%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.52+%2B+%280.88%29%2Ae%5E%28-0.002x%29+=+4 ALIGN=MIDDLE  ALT=\"3.52+%2B+%280.88%29%2Ae%5E%28-0.002x%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting 3.52 from each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280.88%29%2Ae%5E%28-0.002x%29+=+0.48 ALIGN=MIDDLE  ALT=\"%280.88%29%2Ae%5E%28-0.002x%29+=+0.48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide by 0.88:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-0.002x%29+=+%280.88%29%2A0.48 ALIGN=MIDDLE  ALT=\"e%5E%28-0.002x%29+=+%280.88%29%2A0.48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-0.002x%29+=+0.4224 ALIGN=MIDDLE  ALT=\"e%5E%28-0.002x%29+=+0.4224\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now that the exponential part of the equation is isolated, we use logarithms to proceed. Since the base of the exponent is e, it is best to use base e logarithms (aka lm):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%28-0.002x%29%29+=+ln%280.4224%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%28-0.002x%29%29+=+ln%280.4224%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the left side we can use a property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to \"move\" the exponent out in front of the logarithm. (It is this property which is the very reason we use logarithms. It gives us a way to get the variable out of the exponent.) Using the property on the left side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-0.002x%29ln%28e%29+=+ln%280.4224%29 ALIGN=MIDDLE  ALT=\"%28-0.002x%29ln%28e%29+=+ln%280.4224%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By definition, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%29+=+1 ALIGN=MIDDLE  ALT=\"ln%28e%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.002x+=+ln%280.4224%29 ALIGN=MIDDLE  ALT=\"-0.002x+=+ln%280.4224%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Last of all we divide by -0.002:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%280.4224%29%2F%28-0.002%29 ALIGN=MIDDLE  ALT=\"x+=+ln%280.4224%29%2F%28-0.002%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is an exact expression for your answer. Use your calculator if you want a decimal approximation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0.8618025465900853%29%2F%28-0.002%29 ALIGN=MIDDLE  ALT=\"x+=+%28-0.8618025465900853%29%2F%28-0.002%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 430.9012732950426674<BR>\n" );
document.write( "So for a price of $699 the demand will be approximately 431.<br><BR>\n" );
document.write( "For a price of $400, replace the p with 400 and repeat these steps to find the demand at that price. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );