document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190019>Question 1190019</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The total expenditures on benefits​ (in billions of​ dollars) can be approximated by the function h(x)=23.4(1.08)^x ​, where x​ = 5 corresponds to the year 1995.\r<BR>\n" );
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document.write( "​(a) What was the amount for total expenditures in 2011​?<BR>\n" );
document.write( "​(b) What was the first full year in which expenditures exceeded ​$105 ​billion?\r<BR>\n" );
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document.write( "SOLVE A & B PLEASE </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821571 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=821571\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the function is h(x) = 23.4 * 1.08 ^ x<BR>\n" );
document.write( "x = 5 corresponds to the year 1995.<BR>\n" );
document.write( "therefore, x = 0 corresponds to the year 1990, because 1995 - 1990 = 5.<BR>\n" );
document.write( "you have:<BR>\n" );
document.write( "1990 = 0<BR>\n" );
document.write( "1991 = 1<BR>\n" );
document.write( "1992 = 2<BR>\n" );
document.write( "1993 = 3<BR>\n" );
document.write( "1994 = 4<BR>\n" );
document.write( "1995 = 5.<BR>\n" );
document.write( "the value of x is equal to 23.4 because 23.4 * 1.08 ^ 0 = 23.4.<BR>\n" );
document.write( "the value of the investment in 2011 is equal to 23.4 * 1.08 ^ (2011 - 1990) = 23.4 * 1.08 ^ 21 = 117.7917089.<BR>\n" );
document.write( "to find the first year that the expenditures exceeded 105 billion, replace h(x) with 105 to get:<BR>\n" );
document.write( "105 = 23.4 * 1.08 ^ x<BR>\n" );
document.write( "divide both sides of the equation by 24.4 to get:<BR>\n" );
document.write( "105/23.4 = 1.08 ^ x<BR>\n" );
document.write( "take the log of both sides of the equation to get:<BR>\n" );
document.write( "log(105/23.4) = log(1.08 ^ x)<BR>\n" );
document.write( "this becomes log(105/23.4) = x * log(1.08) by the laws of log exponentiation<BR>\n" );
document.write( "divide both sides of the equation by log(1.05) to get:<BR>\n" );
document.write( "log(105/23.4) / log(1.08) = x<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 19.50628923.<BR>\n" );
document.write( "confirm by replacing x in the original equation to get:<BR>\n" );
document.write( "h(x) = 23.4 * 1.08 ^ 19.50628923 = 105.<BR>\n" );
document.write( "the first year that expenditures reach 105 billion is year 19, which would be 1990 + 19 = 2009.<BR>\n" );
document.write( "the actual point is somewhere between 2009 and 2010.<BR>\n" );
document.write( "here's a graph of the equation and the important points.<BR>\n" );
document.write( "x = 0 = 1990<BR>\n" );
document.write( "x = 5 = 1995<BR>\n" );
document.write( "x = 19 = 2009<BR>\n" );
document.write( "x = 20 = 2010<BR>\n" );
document.write( "x = 21 = 2020\r<BR>\n" );
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document.write( "let me know if you have any questions.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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