document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1096702>Question 1096702</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Exponential Models: In a civil case a large company has a judgement made against it.<BR>\n" );
document.write( "The magistrate states that the penalty is $1 million, to be paid on July 01 2017 and then fine <BR>\n" );
document.write( " increases by $10 million each day thereafter. The companys legal counsel insists<BR>\n" );
document.write( "that the penalty is unfair and arbitrary and hence they are not legally bound to pay the fine<BR>\n" );
document.write( " until an appeal case is heard. The magistrate is very canny and in response to this<BR>\n" );
document.write( "they decide to set aside their intial ruling and instead makes the following offer of a choice<BR>\n" );
document.write( "of penalties, with the chosen penalty to be legally binding once agreed to:<BR>\n" );
document.write( "Penalty A: $1 million dollars to be paid on July 1 , 2017 and the ne increases by $10 million<BR>\n" );
document.write( "each day thereafter i.e. the original penalty<BR>\n" );
document.write( "Penalty B: An initial amount of 1 cent, beginning on July 1, 2017 and doubling thereafter, with the ne payable to be the amount generated after 40 days of doubling<BR>\n" );
document.write( "The companylegal counsel is delighted andwithout recourse toaconsulting mathematician,<BR>\n" );
document.write( "readily take the o&#8260;er of Penalty B, thinking that a penalty of 1 cent cannot double su¢ ciently<BR>\n" );
document.write( "to be anywhere near Penalty A.<BR>\n" );
document.write( "1. (a) Which penalty will be larger after the 40 days? Were the company counsel correct<BR>\n" );
document.write( "in taking the option of Penalty B?<BR>\n" );
document.write( "(b) Express as a formula, the amount Penalty A as a function of time t , where t is the<BR>\n" );
document.write( "number of days after July 01 2017.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #711111 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=711111\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Looking at 40 days (which isn't after 40 days but is the boundary).<BR>\n" );
document.write( "Penalty A is 1+(10)(40)=$401 million.  The formula is Cost=$1+$10*t, where the dollars are in millions, and t is the number of days after 1 July ANSWER.<BR>\n" );
document.write( "For penalty B, look at it in cents first.  <BR>\n" );
document.write( "This is the sum (sigma) of 2^t, with t taken from 0 to 40.<BR>\n" );
document.write( "The sum of powers of n: are the highest power (6), add 1 to that power (7) and subtract 1 from the result.\r<BR>\n" );
document.write( "\n" );
document.write( "{The sum of 1+2+4+8+16+32+64 is 127).  Here, we have an initial amount of 1 cent or 2^0.<BR>\n" );
document.write( "Day 1 is 1 or 2^0,  on day 2 the sum is 3 cents (2^2)-1,  day 3, 4+2+1 or 7 cents (2^3-1); day 4, 15 cents (8+4+2+1) or 2^4-1.<BR>\n" );
document.write( "On day 40, it will be 2^40-1 cents.<BR>\n" );
document.write( "That is 1.1  x 10 ^12 cents.  While getting the exact number of cents is more difficult, in this instance we aren't asked for that but the relative costs.<BR>\n" );
document.write( "This is $1.1 x 10^10, or $11 billion.<BR>\n" );
document.write( "A formula in dollars is (1/100)(2^t)-1, where t= days, but this uses 1 July and not the days after which is what the question asks for.  So, for t days after the first, it is 2^(t+1)-2 in cents  (2+4+8+16+32+64+...2^40)<BR>\n" );
document.write( "Penalty A is a far wiser choice. The company made the WRONG choice.  ANSWER </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );