document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174372>Question 1174372</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A vehicle driver has to pay an annual road tax of RM 810 and RM 90 for insurance. His vehicle<BR>\n" );
document.write( "can travel 500 kilometres to one gallon which costs 100 cents per gallon. The vehicle is<BR>\n" );
document.write( "compulsory to be sent for service for every 5 000 kilometres travelled at a cost of RM 1 000, and<BR>\n" );
document.write( "depreciation is calculated in cent by multiplying the square of the mileage by 0.1.<BR>\n" );
document.write( "(a) If he covers x kilometres in a year, obtain an expression for the total cost in travelling x<BR>\n" );
document.write( "kilometres and the average total cost per kilometre. \r<BR>\n" );
document.write( "\n" );
document.write( "(b) Show that the total cost of travelling is RM 1 992 when the average total cost per kilometre<BR>\n" );
document.write( "is minimized. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850694 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=850694\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**(a) Total Cost and Average Total Cost**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Fixed Costs:**<BR>\n" );
document.write( "    * Annual Road Tax: RM 810<BR>\n" );
document.write( "    * Annual Insurance: RM 90<BR>\n" );
document.write( "    * Total Fixed Costs: RM 810 + RM 90 = RM 900\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Fuel Cost:**<BR>\n" );
document.write( "    * Distance per gallon: 500 km<BR>\n" );
document.write( "    * Cost per gallon: 100 cents = RM 1<BR>\n" );
document.write( "    * Number of gallons needed for x km: x / 500<BR>\n" );
document.write( "    * Fuel Cost: (x / 500) * 1 = x / 500 RM\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Service Cost:**<BR>\n" );
document.write( "    * Service every 5000 km: x / 5000 services<BR>\n" );
document.write( "    * Cost per service: RM 1000<BR>\n" );
document.write( "    * Service Cost: (x / 5000) * 1000 = x / 5 RM\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Depreciation Cost:**<BR>\n" );
document.write( "    * Depreciation per km: 0.1 cents = RM 0.001<BR>\n" );
document.write( "    * Total depreciation: 0.001 * x² RM\r<BR>\n" );
document.write( "\n" );
document.write( "5.  **Total Cost (C(x)):**<BR>\n" );
document.write( "    * C(x) = Fixed Costs + Fuel Cost + Service Cost + Depreciation Cost<BR>\n" );
document.write( "    * C(x) = 900 + x / 500 + x / 5 + 0.001x²<BR>\n" );
document.write( "    * C(x) = 900 + 0.002x + 0.2x + 0.001x²<BR>\n" );
document.write( "    * C(x) = 0.001x² + 0.202x + 900\r<BR>\n" );
document.write( "\n" );
document.write( "6.  **Average Total Cost per Kilometer (A(x)):**<BR>\n" );
document.write( "    * A(x) = C(x) / x<BR>\n" );
document.write( "    * A(x) = (0.001x² + 0.202x + 900) / x<BR>\n" );
document.write( "    * A(x) = 0.001x + 0.202 + 900 / x\r<BR>\n" );
document.write( "\n" );
document.write( "**(b) Minimum Average Total Cost**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Find the Derivative of A(x):**<BR>\n" );
document.write( "    * A(x) = 0.001x + 0.202 + 900x⁻¹<BR>\n" );
document.write( "    * A'(x) = 0.001 - 900x⁻²<BR>\n" );
document.write( "    * A'(x) = 0.001 - 900 / x²\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Set A'(x) = 0 to Find Critical Points:**<BR>\n" );
document.write( "    * 0.001 - 900 / x² = 0<BR>\n" );
document.write( "    * 0.001 = 900 / x²<BR>\n" );
document.write( "    * x² = 900 / 0.001<BR>\n" );
document.write( "    * x² = 900000<BR>\n" );
document.write( "    * x = √900000 = 948.68 km\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Find the Minimum Average Total Cost:**<BR>\n" );
document.write( "    * Substitute x = 948.68 into A(x):<BR>\n" );
document.write( "        * A(948.68) = 0.001(948.68) + 0.202 + 900 / 948.68<BR>\n" );
document.write( "        * A(948.68) ≈ 0.94868 + 0.202 + 0.94868<BR>\n" );
document.write( "        * A(948.68) ≈ 2.09936 RM/km\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Find the Total Cost at x = 948.68 km:**<BR>\n" );
document.write( "    * C(948.68) = 0.001(948.68)² + 0.202(948.68) + 900<BR>\n" );
document.write( "    * C(948.68) ≈ 900 + 202 + 900 = 1992 RM\r<BR>\n" );
document.write( "\n" );
document.write( "5.  **Show that it is a Minimum:**<BR>\n" );
document.write( "    * Find the second derivative of A(x):<BR>\n" );
document.write( "        * A'(x) = 0.001 - 900x⁻²<BR>\n" );
document.write( "        * A''(x) = 1800x⁻³ = 1800 / x³<BR>\n" );
document.write( "    * Since x > 0, A''(x) > 0. This means the function is concave up, and we have a minimum.\r<BR>\n" );
document.write( "\n" );
document.write( "**Conclusion:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **(a)** The total cost is C(x) = 0.001x² + 0.202x + 900, and the average total cost is A(x) = 0.001x + 0.202 + 900 / x.<BR>\n" );
document.write( "* **(b)** When the average total cost per kilometer is minimized (at x ≈ 948.68 km), the total cost is approximately RM 1992.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );