document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198754>Question 1198754</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 9)	The demand equation for a manufacturing product is p=20-0.25q, where q is the number of units and p is the price per unit.\r<BR>\n" );
document.write( "\n" );
document.write( "(a) Write the Total Revenue (TR), Average Revenue (AR) and Marginal Revenue (MR) equations. <BR>\n" );
document.write( "(b) At what value of q will there be the total revenue at its maximum? <BR>\n" );
document.write( "(c) What price should the firm charge to achieve this maximum total revenue? <BR>\n" );
document.write( "(d) What is the maximum total revenue?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848273 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848273\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Revenue Equations**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Total Revenue (TR):** <BR>\n" );
document.write( "    * TR = Price (p) * Quantity (q)<BR>\n" );
document.write( "    * TR = (20 - 0.25q) * q <BR>\n" );
document.write( "    * TR = 20q - 0.25q²\r<BR>\n" );
document.write( "\n" );
document.write( "* **Average Revenue (AR):** <BR>\n" );
document.write( "    * AR = Total Revenue (TR) / Quantity (q)<BR>\n" );
document.write( "    * AR = (20q - 0.25q²) / q<BR>\n" );
document.write( "    * AR = 20 - 0.25q \r<BR>\n" );
document.write( "\n" );
document.write( "* **Marginal Revenue (MR):** <BR>\n" );
document.write( "    * MR is the derivative of Total Revenue with respect to quantity (q):<BR>\n" );
document.write( "    * MR = d(TR)/dq = d(20q - 0.25q²)/dq <BR>\n" );
document.write( "    * MR = 20 - 0.5q\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Maximum Total Revenue**\r<BR>\n" );
document.write( "\n" );
document.write( "* Total Revenue is maximized where the derivative of the TR function (MR) is equal to zero.<BR>\n" );
document.write( "    * 20 - 0.5q = 0<BR>\n" );
document.write( "    * 0.5q = 20<BR>\n" );
document.write( "    * q = 40\r<BR>\n" );
document.write( "\n" );
document.write( "* **To confirm this is a maximum:** <BR>\n" );
document.write( "    * The second derivative of the TR function is -0.5, which is negative. This indicates that the point where MR = 0 is indeed a maximum.\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Price to Achieve Maximum Total Revenue**\r<BR>\n" );
document.write( "\n" );
document.write( "* Substitute the value of q (40) into the demand equation:<BR>\n" );
document.write( "    * p = 20 - 0.25 * 40<BR>\n" );
document.write( "    * p = 20 - 10<BR>\n" );
document.write( "    * p = $10\r<BR>\n" );
document.write( "\n" );
document.write( "**d) Maximum Total Revenue**\r<BR>\n" );
document.write( "\n" );
document.write( "* Substitute the value of q (40) into the Total Revenue equation:<BR>\n" );
document.write( "    * TR = 20q - 0.25q² <BR>\n" );
document.write( "    * TR = 20 * 40 - 0.25 * 40²<BR>\n" );
document.write( "    * TR = 800 - 400<BR>\n" );
document.write( "    * TR = $400\r<BR>\n" );
document.write( "\n" );
document.write( "**In Summary:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Total Revenue (TR):** TR = 20q - 0.25q²<BR>\n" );
document.write( "* **Average Revenue (AR):** AR = 20 - 0.25q<BR>\n" );
document.write( "* **Marginal Revenue (MR):** MR = 20 - 0.5q<BR>\n" );
document.write( "* **Quantity for Maximum Total Revenue:** q = 40 units<BR>\n" );
document.write( "* **Price for Maximum Total Revenue:** p = $10<BR>\n" );
document.write( "* **Maximum Total Revenue:** $400<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );