document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168548>Question 1168548</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3. Marginal cost. Suppose the daily cost, in hundreds of <BR>\n" );
document.write( "dollars, of producing x security systems is<BR>\n" );
document.write( "C1x2 = 0.002x3 + 0.1x2 + 42x + 300,<BR>\n" );
document.write( " and currently 40 security systems are produced daily.<BR>\n" );
document.write( "a) What is the current daily cost?<BR>\n" );
document.write( "b) What would be the additional daily cost of increasing <BR>\n" );
document.write( "production to 41 security systems daily?<BR>\n" );
document.write( "c) What is the marginal cost when x = 40?<BR>\n" );
document.write( "d) Use marginal cost to estimate the daily cost of increasing <BR>\n" );
document.write( "production to 42 security systems daily. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851560 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=851560\">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" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Cost function: C(x) = 0.002x³ + 0.1x² + 42x + 300 (where C(x) is in hundreds of dollars)<BR>\n" );
document.write( "* Current production: x = 40 security systems\r<BR>\n" );
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document.write( "**a) What is the current daily cost?**\r<BR>\n" );
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document.write( "* C(40) = 0.002(40)³ + 0.1(40)² + 42(40) + 300<BR>\n" );
document.write( "* C(40) = 0.002(64000) + 0.1(1600) + 1680 + 300<BR>\n" );
document.write( "* C(40) = 128 + 160 + 1680 + 300<BR>\n" );
document.write( "* C(40) = 2268\r<BR>\n" );
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document.write( "Since C(x) is in hundreds of dollars, the current daily cost is 2268 * 100 = $226,800.\r<BR>\n" );
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document.write( "**b) What would be the additional daily cost of increasing production to 41 security systems daily?**\r<BR>\n" );
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document.write( "* C(41) = 0.002(41)³ + 0.1(41)² + 42(41) + 300<BR>\n" );
document.write( "* C(41) = 0.002(68921) + 0.1(1681) + 1722 + 300<BR>\n" );
document.write( "* C(41) = 137.842 + 168.1 + 1722 + 300<BR>\n" );
document.write( "* C(41) = 2327.942\r<BR>\n" );
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document.write( "* Additional cost = C(41) - C(40)<BR>\n" );
document.write( "* Additional cost = 2327.942 - 2268 = 59.942\r<BR>\n" );
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document.write( "Since C(x) is in hundreds of dollars, the additional cost is 59.942 * 100 = $5,994.20.\r<BR>\n" );
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document.write( "**c) What is the marginal cost when x = 40?**\r<BR>\n" );
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document.write( "* Marginal cost is the derivative of the cost function, C'(x).<BR>\n" );
document.write( "* C'(x) = 0.006x² + 0.2x + 42\r<BR>\n" );
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document.write( "* C'(40) = 0.006(40)² + 0.2(40) + 42<BR>\n" );
document.write( "* C'(40) = 0.006(1600) + 8 + 42<BR>\n" );
document.write( "* C'(40) = 9.6 + 8 + 42<BR>\n" );
document.write( "* C'(40) = 59.6\r<BR>\n" );
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document.write( "Since C(x) is in hundreds of dollars, the marginal cost at x = 40 is 59.6 * 100 = $5,960.\r<BR>\n" );
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document.write( "**d) Use marginal cost to estimate the daily cost of increasing production to 42 security systems daily.**\r<BR>\n" );
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document.write( "* Marginal cost at x = 40 gives an estimate for the cost of producing one additional unit.<BR>\n" );
document.write( "* To estimate the cost of producing two additional units, we can multiply the marginal cost at x = 40 by 2.\r<BR>\n" );
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document.write( "* Estimated additional cost for 2 units = 2 * C'(40)<BR>\n" );
document.write( "* Estimated additional cost for 2 units = 2 * 59.6 = 119.2\r<BR>\n" );
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document.write( "Since C(x) is in hundreds of dollars, the estimated additional cost is 119.2 * 100 = $11,920.\r<BR>\n" );
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document.write( "* Estimated cost of producing 42 units:<BR>\n" );
document.write( "    * C(42) = 0.002(42)^3 + 0.1(42)^2 + 42(42) + 300<BR>\n" );
document.write( "    * C(42) = 0.002(74088) + 0.1(1764) + 1764 + 300<BR>\n" );
document.write( "    * C(42) = 148.176 + 176.4 + 1764 + 300<BR>\n" );
document.write( "    * C(42) = 2388.576<BR>\n" );
document.write( "* C(42) = 2388.576 * 100 = $238,857.60\r<BR>\n" );
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document.write( "* C(42) - C(40) = 2388.576 - 2268 = 120.576<BR>\n" );
document.write( "* 120.576 * 100 = $12,057.60\r<BR>\n" );
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document.write( "The marginal cost estimation of $11,920 is very close to the actual additional cost of $12,057.60.\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "a) $226,800<BR>\n" );
document.write( "b) $5,994.20<BR>\n" );
document.write( "c) $5,960<BR>\n" );
document.write( "d) $11,920 (estimated additional cost)<BR>\n" );
document.write( " </SPAN>\n" );
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