document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1110574>Question 1110574</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A thermometer reading 19 Fahrenheit is brought into a room whose temperature is 65 F. After a minute, the thermometer reading is 30 F. • Set up an equation for predicting the thermometer reading at any time t in minutes. • Find the temperature 4 minutes after the thermometer is brought into the room. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #725702 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=725702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The initial temperature difference in degrees Fahrenheit is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65-19=46 ALIGN=MIDDLE  ALT=\"65-19=46\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Nature wants to get to equilibrium,<BR>\n" );
document.write( "and it heads there at a rate<BR>\n" );
document.write( "proportional to the magnitude of the deviation from equilibrium.<BR>\n" );
document.write( "Without talking about differential equations,<BR>\n" );
document.write( "the difference in temperature will undergo \"exponential decay.\"<BR>\n" );
document.write( "The temperature difference will be an exponential function of time.<BR>\n" );
document.write( "Because it has been found so useful for calculus,<BR>\n" );
document.write( "we like to write exponential functions using the irrational number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >=time in minutes since the thermometer was brought into a room,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > being the thermometer reading in degrees Fahrenheit,<BR>\n" );
document.write( "the temperature difference function will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65-F=46%2Ae%5E%28-kt%29 ALIGN=MIDDLE  ALT=\"65-F=46%2Ae%5E%28-kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with a positive constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be determined.<BR>\n" );
document.write( "If we want, we can solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=65-46%2Ae%5E%28-kt%29 ALIGN=MIDDLE  ALT=\"F=65-46%2Ae%5E%28-kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "To determine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > we substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=30 ALIGN=MIDDLE  ALT=\"F=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65-F=46%2Ae%5E%28-kt%29 ALIGN=MIDDLE  ALT=\"65-F=46%2Ae%5E%28-kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65-30=46%2Ae%5E%28-k%29 ALIGN=MIDDLE  ALT=\"65-30=46%2Ae%5E%28-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35=46%2Ae%5E%28-k%29 ALIGN=MIDDLE  ALT=\"35=46%2Ae%5E%28-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2F46=e%5E%28-k%29 ALIGN=MIDDLE  ALT=\"35%2F46=e%5E%28-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2835%2F46%29=-k ALIGN=MIDDLE  ALT=\"ln%2835%2F46%29=-k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-ln%2835%2F46%29=about0.27329 ALIGN=MIDDLE  ALT=\"k=-ln%2835%2F46%29=about0.27329\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounding to 5 decimal places).<BR>\n" );
document.write( " <BR>\n" );
document.write( "So, the thermometer reading is predicted by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28F=65-46%2Ae%5E%28-0.27329t%29%29 ALIGN=MIDDLE  ALT=\"highlight%28F=65-46%2Ae%5E%28-0.27329t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Four (4) minutes after the thermometer is brought into the room, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4 ALIGN=MIDDLE  ALT=\"t=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=65-46%2Ae%5E%28-0.27329%2A4%29=65-46%2Ae%5E%28-1.09316%29=65-46%2A0.33516=65-15.4=49.6 ALIGN=MIDDLE  ALT=\"F=65-46%2Ae%5E%28-0.27329%2A4%29=65-46%2Ae%5E%28-1.09316%29=65-46%2A0.33516=65-15.4=49.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The temperature reading 4 minutes after the thermometer is brought into the room<BR>\n" );
document.write( "is expected to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2850%29 ALIGN=MIDDLE  ALT=\"highlight%2850%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > degrees Fahrenheit. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );