document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189396>Question 1189396</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A car engine runs at a temperature of 87°C, when the engine is turned off, it cools according to Newton’s Law of Cooling (T=Ts+D0e-kt) with constant k=0.0347, where the time is measured in minutes. How long does it take for the engine to cool to 33°C if the surrounding temperature is 15°C. (A) 44min (B) 42min (C) 40min (D) 38min (E) 36min (note: ln2=0.6931)<BR>\n" );
document.write( "How can i solve this without a calculator? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820747 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=820747\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the engine is 87 degrees centigrade when it is shut off.\r<BR>\n" );
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document.write( "newton's law of cooling is:\r<BR>\n" );
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document.write( "T(t) = Ts + (T0 - Ts) * e ^ -kt.\r<BR>\n" );
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document.write( "T(t) is equal to 33 degrees.<BR>\n" );
document.write( "T0 is equal to 87 degrees.<BR>\n" );
document.write( "Ts is equal to 15 degrees.\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "33 = 15 + (87 - 15) * e ^ (-.0347 * t)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "33 = 15 + 72 * e ^ (-.0347 * t)\r<BR>\n" );
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document.write( "subtract 15 from both sides of the equation to get:\r<BR>\n" );
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document.write( "18 = 72 * e ^ (-.0347 * t)\r<BR>\n" );
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document.write( "divide both sides of the equation by 72 to get:\r<BR>\n" );
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document.write( "18/72 = e ^ (-.0347 * t)\r<BR>\n" );
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document.write( "take the natural log of both sides of the equation to get:\r<BR>\n" );
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document.write( "ln(18/72) = ln(e ^ (-.0347 * t)\r<BR>\n" );
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document.write( "by laws of logs, this becomes:\r<BR>\n" );
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document.write( "ln(18/72) = -.0347 * t * ln(e).\r<BR>\n" );
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document.write( "since ln(e) = 1, this becomes:\r<BR>\n" );
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document.write( "ln(18/72) = = -.0347 * t\r<BR>\n" );
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document.write( "divide both sides of the equation by -.0347 to get:\r<BR>\n" );
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document.write( "ln(18/72) / -.0347 = t\r<BR>\n" );
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document.write( "solve for t to get:\r<BR>\n" );
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document.write( "t = 39.95084614.\r<BR>\n" );
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document.write( "since since 39.95... rounded to 40, your solution is that t = 40 minutes.\r<BR>\n" );
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document.write( "the online calculator at <a  rel=nofollow HREF= \"https://www.calctown.com/calculators/newton-law-of-cooling\" target = \"_blank\">https://www.calctown.com/calculators/newton-law-of-cooling</a> give the same result.\r<BR>\n" );
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document.write( "here's a display of the results from using that calculator.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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