document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039863>Question 1039863</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I would appreciate any help that can be offered on this problem: \r<BR>\n" );
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document.write( "EXPONENTIAL REGRESSION<BR>\n" );
document.write( "Data: A cup of hot coffee was placed in a room maintained at a constant temperature of 72 degrees, and the coffee temperature was recorded periodically, in Table 1. \r<BR>\n" );
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document.write( "t = Time Elapsed<BR>\n" );
document.write( "(minutes)	C = Coffee<BR>\n" );
document.write( "Temperature (degrees F.)<BR>\n" );
document.write( "0	169.0<BR>\n" );
document.write( "10	143.5<BR>\n" );
document.write( "20	128.2<BR>\n" );
document.write( "30	113.3<BR>\n" );
document.write( "40	107.5<BR>\n" );
document.write( "50	101.4<BR>\n" );
document.write( "60	96.9<BR>\n" );
document.write( "TABLE 1	REMARKS: <BR>\n" );
document.write( "Common sense tells us that the coffee will be cooling off and its temperature will decrease and approach the ambient temperature of the room, 72 degrees. \r<BR>\n" );
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document.write( "So, the temperature difference between the coffee temperature and the room temperature will decrease to 0.\r<BR>\n" );
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document.write( "We will fit the temperature difference data (Table 2) to an exponential curve of the form y = A e^-bt. \r<BR>\n" );
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document.write( "Notice that as t gets large, y will get closer and closer to 0, which is what the temperature difference will do. <BR>\n" );
document.write( "So, we want to analyze the data where t = time elapsed and y = C - 72, the temperature difference between the coffee temperature and the room temperature.	\r<BR>\n" );
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document.write( "TABLE 2<BR>\n" );
document.write( "t = Time Elapsed (minutes)	y = C - 72 Temperature <BR>\n" );
document.write( "Difference<BR>\n" );
document.write( "(degrees F.)<BR>\n" );
document.write( "0	97.0<BR>\n" );
document.write( "10	71.5<BR>\n" );
document.write( "20	56.2<BR>\n" );
document.write( "30	41.3<BR>\n" );
document.write( "40	35.5<BR>\n" );
document.write( "50	29.4<BR>\n" );
document.write( "60	24.9 \r<BR>\n" );
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document.write( "Exponential Function of Best Fit (using the data in Table 2):<BR>\n" );
document.write( "y = 89.976 e^-.023t where t = Time Elapsed (minutes) and y = Temperature Difference (in degrees) \r<BR>\n" );
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document.write( "(a) Use the exponential function to estimate the temperature difference y when 15 minutes have elapsed. Report your estimated temperature difference to the nearest tenth of a degree. \r<BR>\n" );
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document.write( "(b) Since y = C - 72, we have coffee temperature C = y + 72. Take your difference estimate from part (a) and add 72 degrees. Interpret the result by filling in the blank:\r<BR>\n" );
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document.write( "When 15 minutes have elapsed, the estimated coffee temperature is ________ degrees. \r<BR>\n" );
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document.write( "(c) Suppose the coffee temperature C is 105 degrees. Then y = C - 72 = ____ degrees is the temperature difference between the coffee and room temperatures. \r<BR>\n" );
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document.write( "(d) Consider the equation _____ = 89.976 e^-.023t where the ____ is filled in with your answer from part (c). \r<BR>\n" );
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document.write( "I really appreciate any help that can be offered on this, thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654709 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=654709\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "a.  Substitute 15 for t in the exponential equation and then do the indicated arithmetic.  Given the precision shown in the table, you can probably round to the nearest 10th of a degree.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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