document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138844>Question 1138844</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help with this question with exponential functions <BR>\n" );
document.write( "A cup of hot liquid was left to cool in a room whose temperature was 18°C. The temperature changes according to the function T(t)=80(1/2)^t/30+18 What are the y-intercept and the equation of the horizontal asymptote?  What do they mean in this context?<BR>\n" );
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document.write( "The equation you show for the temperature is not correct:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28t%29=80%281%2F2%29%5Et%2F30%2B18 ALIGN=MIDDLE  ALT=\"T%28t%29=80%281%2F2%29%5Et%2F30%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Assuming you know something about how a hot liquid cools, you know this is not the right equation.  You should know that parentheses are required:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28t%29=80%281%2F2%29%5E%28t%2F30%29%2B18 ALIGN=MIDDLE  ALT=\"T%28t%29=80%281%2F2%29%5E%28t%2F30%29%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "As with any function, the y-intercept is the value when x=0.  For this equation,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%280%29+=+80%281%2F2%29%5E%280%2F30%29%2B18+=+80%281%29%2B18+=+98 ALIGN=MIDDLE  ALT=\"T%280%29+=+80%281%2F2%29%5E%280%2F30%29%2B18+=+80%281%29%2B18+=+98\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The horizontal asymptote is the value of the function when t gets very large.  For this equation, since 1/2 to a very large power approaches 0,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28infinity%29+=+0%2B18+=+18 ALIGN=MIDDLE  ALT=\"T%28infinity%29+=+0%2B18+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this problem, it should be clear that the y-intercept 98 is the initial temperature of the liquid (in °C).<br><BR>\n" );
document.write( "The horizontal asymptote is 18, which is the temperature in °C of the room.  That makes sense; a hot liquid left in a room for a very long time will eventually have a temperature that is the same as the temperature of the room.<br><BR>\n" );
document.write( "Note that, while the problem doesn't ask anything about it, the exponent \"t/30\" means that the difference between the temperature of the liquid and the room temperature gets cut in half every 30 units of time (presumably minutes). </SPAN>\n" );
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