document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195967>Question 1195967</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An appliance service company is located centrally in roughly square area x miles on a side. It charges $27 per call, not including parts and labor, and travel cost is figured at $1.50 per mile. The average distance traveled per call is 1.2x miles. In a month the average number of calls per square mile of service area is 30.\r<BR>\n" );
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document.write( "a)What should x be if net travel income (which excludes parts and labor) is to be maximized?<BR>\n" );
document.write( "b)What lot size yields minimum cost?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848497 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848497\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Determine the Optimal Service Area Size (x) for Maximum Net Travel Income**\r<BR>\n" );
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document.write( "* **Net Travel Income:** <BR>\n" );
document.write( "    * Net Travel Income = (Revenue per call - Travel cost per call) * (Total calls)<BR>\n" );
document.write( "    * Net Travel Income = ($27 - $1.80x) * (30x²) <BR>\n" );
document.write( "    * Net Travel Income = 810x² - 54x³\r<BR>\n" );
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document.write( "* **Find the derivative of the Net Travel Income function:**<BR>\n" );
document.write( "    * d(Net Travel Income)/dx = 1620x - 162x² \r<BR>\n" );
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document.write( "* **Set the derivative equal to zero to find critical points:**<BR>\n" );
document.write( "    * 1620x - 162x² = 0<BR>\n" );
document.write( "    * 162x(10 - x) = 0\r<BR>\n" );
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document.write( "* **Solve for x:**<BR>\n" );
document.write( "    * x = 0 (This is a trivial solution)<BR>\n" );
document.write( "    * x = 10 \r<BR>\n" );
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document.write( "* **To verify that x = 10 maximizes net travel income:**<BR>\n" );
document.write( "    * **Second Derivative Test:**<BR>\n" );
document.write( "        * d²(Net Travel Income)/dx² = 1620 - 324x<BR>\n" );
document.write( "        * At x = 10, d²(Net Travel Income)/dx² = 1620 - 324(10) = -1620 <BR>\n" );
document.write( "        * Since the second derivative is negative at x = 10, it indicates a maximum.\r<BR>\n" );
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document.write( "* **Therefore, to maximize net travel income, the service area should have a side length (x) of 10 miles.**\r<BR>\n" );
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document.write( "**b) Finding the Lot Size for Minimum Cost**\r<BR>\n" );
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document.write( "* **The question about \"lot size\" is not directly related to the given information about the service area.** <BR>\n" );
document.write( "* Lot size typically refers to the quantity of goods ordered or produced in a single production run. \r<BR>\n" );
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document.write( "* **To determine the lot size that yields minimum cost, you would need additional information:**<BR>\n" );
document.write( "    * **Ordering costs:** Costs associated with placing an order (e.g., administrative costs, transportation costs).<BR>\n" );
document.write( "    * **Holding costs:** Costs associated with storing inventory (e.g., storage space, insurance, spoilage).<BR>\n" );
document.write( "    * **Demand:** The rate at which the goods are used.\r<BR>\n" );
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document.write( "* **Common inventory management models (like the Economic Order Quantity - EOQ) can be used to determine the optimal lot size that minimizes the sum of ordering and holding costs.**\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* To maximize net travel income, the service area should have a side length of 10 miles.<BR>\n" );
document.write( "* The question about lot size requires additional information to be answered.\r<BR>\n" );
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document.write( "I hope this comprehensive explanation is helpful!<BR>\n" );
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