document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172102>Question 172102</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.	A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet? \r<BR>\n" );
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document.write( "2.	A business invests $10,000 in a savings account for two years.  At the beginning of the second year, an additional $3500 is invested.  At the end of the second year, the account balance is $15,569.75.  What was the annual interest rate? \r<BR>\n" );
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document.write( "3.	Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle. \r<BR>\n" );
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document.write( "4.	The Hudson River flows at a rate of 3 miles per hour.  A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours.  What is the speed of the boat in still water?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127132 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=127132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> #1:  Let the width of the path be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the overall dimensions of the garden plus the path must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2B2x ALIGN=MIDDLE  ALT=\"18%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2B2x ALIGN=MIDDLE  ALT=\"13%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > because there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > width of path on each of the four sides.  Since the area of the garden by itself is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2A13=234 ALIGN=MIDDLE  ALT=\"18%2A13=234\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the overall area of the garden plus the path must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=234%2B516=750 ALIGN=MIDDLE  ALT=\"234%2B516=750\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But the overall area must also be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2818%2B2x%29%2813%2B2x%29 ALIGN=MIDDLE  ALT=\"%2818%2B2x%29%2813%2B2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2818%2B2x%29%2813%2B2x%29=750 ALIGN=MIDDLE  ALT=\"%2818%2B2x%29%2813%2B2x%29=750\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Once you multiply the binomials and put the constant term on the left, you will have a quadratic equation in x, which is the value we want to find.  You will get two roots to this equation, one of which will be negative.  You can exclude the negative root as extraneous introduced by the process of squaring the variable (negative dimensions are absurd anyway).  The remain root will be your answer.\r<BR>\n" );
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document.write( "2.   The IRR function in Excel with the arguments 10,000, 3,500, and -15569.75 produces a result of 8.5% Checking this result:\r<BR>\n" );
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document.write( "10,000 * 1.085 = 10,850\r<BR>\n" );
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document.write( "(10,850 + 3,500) * 1.085 = 15,569.75  Answer checks\r<BR>\n" );
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document.write( "IRR is an iterative calculation based on a collection of pairs of cash flows and time.  Use the Net Present Value formula set to zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NPV=sum%28C%5Bt%5D%2F%28%281%2Br%29%5Et%29%2Ct=0%2CN%29=0 ALIGN=MIDDLE  ALT=\"NPV=sum%28C%5Bt%5D%2F%28%281%2Br%29%5Et%29%2Ct=0%2CN%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In this case:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NPV=10000%2B3500%2F%28%281%2Br%29%5E1%29-15569.75%2F%28%281%2Br%29%5E2%29=0 ALIGN=MIDDLE  ALT=\"NPV=10000%2B3500%2F%28%281%2Br%29%5E1%29-15569.75%2F%28%281%2Br%29%5E2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for r.\r<BR>\n" );
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document.write( "3.  Distance = 200 miles.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=rt ALIGN=MIDDLE  ALT=\"d=rt\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the distance, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the speed for the given situation, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the time for the given situation, hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200=rt ALIGN=MIDDLE  ALT=\"200=rt\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  In the \"what if\" situation, the speed is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2B10 ALIGN=MIDDLE  ALT=\"r%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the time is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-1 ALIGN=MIDDLE  ALT=\"t-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200+=+%28r%2B10%29%28t-1%29 ALIGN=MIDDLE  ALT=\"200+=+%28r%2B10%29%28t-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Solve both equations for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=200%2Fr ALIGN=MIDDLE  ALT=\"t=200%2Fr\"   DISABLED_style= \"max-width:90%; height:auto;\" > and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28200%2F%28r%2B10%29%29%2B1 ALIGN=MIDDLE  ALT=\"t=%28200%2F%28r%2B10%29%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=t ALIGN=MIDDLE  ALT=\"t=t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200%2Fr=%28200%2F%28r%2B10%29%29%2B1 ALIGN=MIDDLE  ALT=\"200%2Fr=%28200%2F%28r%2B10%29%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Applying the lowest common denominator of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2%2B10r ALIGN=MIDDLE  ALT=\"r%5E2%2B10r\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28200%28r%2B10%29%29%2F%28r%5E2%2B10r%29=%28200r%2Br%5E2%2B10r%29%2F%28r%5E2%2B10r%29 ALIGN=MIDDLE  ALT=\"%28200%28r%2B10%29%29%2F%28r%5E2%2B10r%29=%28200r%2Br%5E2%2B10r%29%2F%28r%5E2%2B10r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplifying and setting the numerators equal:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200r%2B2000=200r%2Br%5E2%2B10r ALIGN=MIDDLE  ALT=\"200r%2B2000=200r%2Br%5E2%2B10r\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Collecting like terms and putting in standard form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2%2B10r-2000=0 ALIGN=MIDDLE  ALT=\"r%5E2%2B10r-2000=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2A-40=-2000 ALIGN=MIDDLE  ALT=\"50%2A-40=-2000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50-40=10 ALIGN=MIDDLE  ALT=\"50-40=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the quadratic factors:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B50%29%28r-40%29=0 ALIGN=MIDDLE  ALT=\"%28r%2B50%29%28r-40%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=-50 ALIGN=MIDDLE  ALT=\"r=-50\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=40 ALIGN=MIDDLE  ALT=\"r=40\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since, presumably, he wasn't going backwards at any time, the -50 root is extraneous.  Therefore the speed of the given trip was 40 mph.\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "200/40 = 5 hours for the trip\r<BR>\n" );
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document.write( "200/50 = 4 hours for the 'what if' trip\r<BR>\n" );
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document.write( "50 is 10 more than 40 and 4 is 1 less than 5, Answer checks.\r<BR>\n" );
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document.write( "4.  This is really the same problem as the previous one.  The speed of the current subtracts from the speed in still water on the upstream trip and adds on the downstream trip.  Call the speed in still water <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the time to go upstream <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since the time to go upstream plus the time to go downstream is 9 hours, the time to go downstream can be expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-t ALIGN=MIDDLE  ALT=\"9-t\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-3%29t=60 ALIGN=MIDDLE  ALT=\"%28r-3%29t=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B3%29%289-t%29=60 ALIGN=MIDDLE  ALT=\"%28r%2B3%29%289-t%29=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve both equations for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > and set the resulting right-hand expressions equal to each other.  When you simplify by using the LCD, collecting like terms, and putting it in standard form, you will have a quadratic in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Solve by factoring or using the quadratic formula as is appropriate.  You will most likely have an extraneous root because of squaring the variable in the process of setting up the equation.  The positive root will be the value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > that answers the question. </SPAN>\n" );
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