document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194378>Question 1194378</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that an outbreak of cholera follows severe f;ooding in an isolated town of 2621 people, initially (Day0), 71 people are infected. Every day after, 12% of thoes still healthy fall ill.\r<BR>\n" );
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document.write( "By what day will 95% of the population be infected? (more specifically on the day before, fewer than this number are infected, but on this day, the number or more are infected)<BR>\n" );
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document.write( "a = initial number of people infected = 71<BR>\n" );
document.write( "r = 0.12 = growth rate in decimal form<BR>\n" );
document.write( "b = 1 + r = 1+0.12 = 1.12 = growth factor\r<BR>\n" );
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document.write( "The template<BR>\n" );
document.write( "y = a*b^x<BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "y = 71*(1.12)^x<BR>\n" );
document.write( "and this is an exponential model for the number who fall ill\r<BR>\n" );
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document.write( "x = number of days<BR>\n" );
document.write( "y = number who are ill\r<BR>\n" );
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document.write( "The town has a population of 2621<BR>\n" );
document.write( "95% of which is 0.95*2621 = 2489.95 which I'll round to the nearest whole number to get 2490\r<BR>\n" );
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document.write( "Once the town has 2490 ill people, it will reach an infection rate of about 95%.\r<BR>\n" );
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document.write( "Replace y with 2490 and solve for x.<BR>\n" );
document.write( "You'll need logarithms to isolate the exponent.\r<BR>\n" );
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document.write( "y = 71*(1.12)^x<BR>\n" );
document.write( "2490 = 71*(1.12)^x<BR>\n" );
document.write( "2490/71 = (1.12)^x<BR>\n" );
document.write( "(1.12)^x = 2490/71<BR>\n" );
document.write( "(1.12)^x = 35.070423<BR>\n" );
document.write( "Log[ (1.12)^x ] = Log[ 35.070423 ]<BR>\n" );
document.write( "x*Log[ 1.12 ] = Log[ 35.070423 ]<BR>\n" );
document.write( "x = Log[ 35.070423 ]/Log[ 1.12 ]<BR>\n" );
document.write( "x = 31.389741\r<BR>\n" );
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document.write( "Let's check to see what happens when we plug in x = 31<BR>\n" );
document.write( "y = 71*(1.12)^x<BR>\n" );
document.write( "y = 71*(1.12)^31<BR>\n" );
document.write( "y = 2382.413007<BR>\n" );
document.write( "y = 2382<BR>\n" );
document.write( "We don't reach the goal of 2490, but we get pretty close.\r<BR>\n" );
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document.write( "Now try x = 32<BR>\n" );
document.write( "y = 71*(1.12)^x<BR>\n" );
document.write( "y = 71*(1.12)^32<BR>\n" );
document.write( "y = 2668.302568<BR>\n" );
document.write( "y = 2668<BR>\n" );
document.write( "We're now over the hurdle of 2490\r<BR>\n" );
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document.write( "Answer: <font color=red>Day 32</font><BR>\n" );
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