document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109448>Question 1109448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Jessica and Matthew work for different companies. Jessica has a base salary of $36,000 and will receive an increase of salary at a rate of 3.1% per year. Matthew has a base salary of $38,000 and will receive an increase of salary at 2.5% per year. Jessica and Matthew are calculating the number of years, t, it will take each of them to earn over $45,000 at their respective companies. Determine the inequality could be used to determine the number of years for each person to reach $45,000. Then, decide which statement is true for each person. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724451 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=724451\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> jessica receives 36000 and gets an increase of 3.1% per year.\r<BR>\n" );
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document.write( "matthew has a base salary of 38000 and gets an increase of 2.5% per year.\r<BR>\n" );
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document.write( "inequality to use is f < p * (1 + r) ^ t.\r<BR>\n" );
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document.write( "f is the future value.<BR>\n" );
document.write( "p is the present value.<BR>\n" );
document.write( "r is the annual interest rate, which is the annual interest rate percent divided by 100.<BR>\n" );
document.write( "t is the number of years.\r<BR>\n" );
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document.write( "for jessica, the inequality becomes 45000 < 36000 * (1 + .031) ^ t\r<BR>\n" );
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document.write( "for matthew, the inequality becomes 45000 < 38000 * (1 + .025) ^ t\r<BR>\n" );
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document.write( "solving for jessica:\r<BR>\n" );
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document.write( "start with 45000 < 36000 * (1 + .031) ^ t\r<BR>\n" );
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document.write( "divide both sides of the inequality by 36000 to get:\r<BR>\n" );
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document.write( "45000 / 36000 < (1 + .031) ^ t\r<BR>\n" );
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document.write( "take the log of both sides of the inequality to get:\r<BR>\n" );
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document.write( "log (45000 / 36000) < log((1 + .031) ^ t)\r<BR>\n" );
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document.write( "since log ((1 + .031) ^ t) is equal to t * log(1 + .031), the equation becomes:\r<BR>\n" );
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document.write( "log(45000 / 36000) < t * log(1 + .031)\r<BR>\n" );
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document.write( "divide both sides of the equation by log(1 + .031) to get:\r<BR>\n" );
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document.write( "log(45000 / 36000) / log(1 + .031) < t\r<BR>\n" );
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document.write( "solve for t to get 7.30918316 < t.\r<BR>\n" );
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document.write( "this is the same as t > 7.30918316.\r<BR>\n" );
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document.write( "when t is equal to that, you get f = 36000 * (1 + .031) ^ 7.30918316.\r<BR>\n" );
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document.write( "this results in f = 45000.\r<BR>\n" );
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document.write( "when t is greater than that, like when t = 7.302 years, you get f = 36000 * (1 + .031) ^ 7.302 = 45074.91875.\r<BR>\n" );
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document.write( "since 45000 < 45074.91875, the inequality is true when t is greater than 7.30918316.\r<BR>\n" );
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document.write( "when you do the same process for matthew, you get:\r<BR>\n" );
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document.write( "45000 < 38000 * (1 + .015) ^ t when t > 6.847243459\r<BR>\n" );
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document.write( "when t is equal to that, you get f = 38000 * (1 + .015) ^ 6.847243459 which results in f = 45000.\r<BR>\n" );
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document.write( "when t is greater than that, like when t = 6.848, you get f = 38000 * (1 + .025) ^ 6.848 = 45000.84065.\r<BR>\n" );
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document.write( "since 45000 < 45000.84065, the inequality is true when t is greater than 6.847243459.\r<BR>\n" );
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document.write( "your solution is that:\r<BR>\n" );
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document.write( "the inequality for jessica of 45000 < 36000 * 1.031 ^ t is true when t > 7.30918336.\r<BR>\n" );
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document.write( "the inequality for matthew of 45000 < 38000 * 1.025 ^ t is true when t > 6.847243459.\r<BR>\n" );
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document.write( "round to as many decimal places as required.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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