document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116301>Question 116301</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131915053/algebrahomeworkh>Elementary & Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> A pleasure boat carries passengers 10 miles upstream and then returns to the starting point. The total time of the trip(excluding the time on the ground) takes 2 hours. If the speed of the current is 2 miles per hour, find the speed of the boat in still water. Round your answer to the nearest tenth of a mile. \r<BR>\n" );
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document.write( "I need to see how you work this problem out in detail! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84573 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=84573\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The basic idea we will use in solving this problem is the formula that says: Distance equals<BR>\n" );
document.write( "speed times time. [You can get a feel for this equation if you think, \"If I go at 60 miles per<BR>\n" );
document.write( "hour for 2 hours I should be 120 miles (60 * 2) from where I started.] Anyhow, we  can write<BR>\n" );
document.write( "this distance equation as:<BR>\n" );
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document.write( "D = S * T<BR>\n" );
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document.write( "where D represents distance, S represents speed, and T represents time.<BR>\n" );
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document.write( "The first sentence of the problem tells you that you are going to travel a distance of<BR>\n" );
document.write( "10 miles upstream. Traveling upstream means that you are going against the direction that<BR>\n" );
document.write( "the stream is flowing, and the actual speed of the boat is its speed in still water (call it<BR>\n" );
document.write( "S) minus the speed of the current (call that s and the problem says it is 2 miles per hour).<BR>\n" );
document.write( "So when you are going upstream for 10 miles at a speed of (S - 2) miles per hour our distance<BR>\n" );
document.write( "equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+%28S+-+2%29%2A+T ALIGN=MIDDLE  ALT=\"10+=+%28S+-+2%29%2A+T\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve this equation for T by dividing both sides of this equation by (S - 2) to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+10%2F%28S+-+2%29 ALIGN=MIDDLE  ALT=\"T+=+10%2F%28S+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This T is the time it takes to go upstream for 10 miles.<BR>\n" );
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document.write( "Now let's look at the return trip. Since you are going to return to the starting point, you will<BR>\n" );
document.write( "again go 10 miles. But this time you will be going in the same direction as the current is<BR>\n" );
document.write( "flowing. Therefore, in this direction the speed of the boat will be its speed in still water<BR>\n" );
document.write( "PLUS the speed of the current. Therefore, the actual speed of the boat is given by (S + 2)<BR>\n" );
document.write( "when you go downstream. And the distance equation for this part of the trip is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+%28S+%2B+2%29%2A+T ALIGN=MIDDLE  ALT=\"10+=+%28S+%2B+2%29%2A+T\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve this equation for the time it takes to go downstream by dividing both sides of the<BR>\n" );
document.write( "equation by (S + 2) to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+10%2F%28S+%2B+2%29 ALIGN=MIDDLE  ALT=\"T+=+10%2F%28S+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally, the problem tells you that the total time of the trip is 2 hours. That means that the<BR>\n" );
document.write( "time going upstream ... for which we have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+10%2F%28S+-+2%29 ALIGN=MIDDLE  ALT=\"T+=+10%2F%28S+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "plus the time to go downstream ... for which we have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+10%2F%28S+%2B+2%29 ALIGN=MIDDLE  ALT=\"T+=+10%2F%28S+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "must add together to give 2 hours. In equation form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+10%2F%28S+-+2%29+%2B+10%2F%28S+%2B+2%29 ALIGN=MIDDLE  ALT=\"2+=+10%2F%28S+-+2%29+%2B+10%2F%28S+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can get rid of the denominators by multiplying this entire equation (all terms on<BR>\n" );
document.write( "both sides by (S - 2)* (S + 2). When you do that the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28S-2%29%2A%28S+%2B+2%29+=+%2810%2A%28S-2%29%2A%28S+%2B+2%29%29%2F%28S+-+2%29+%2B+10%2A%28%28S-2%29%2A%28S+%2B+2%29%29%2F%28S+%2B+2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Cancel the terms in the denominators with their corresponding like terms in the numerators as<BR>\n" );
document.write( "shown:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28S-2%29%2A%28S+%2B+2%29+=+%2810%2Across%28S-2%29%2A%28S+%2B+2%29%29%2Fcross%28S+-+2%29+%2B+10%2A%28%28S-2%29%2Across%28S+%2B+2%29%29%2Fcross%28S+%2B+2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You are then left with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28S-2%29%2A%28S+%2B+2%29+=+10%2A%28S%2B2%29+%2B+10%2A%28S+-+2%29 ALIGN=MIDDLE  ALT=\"2%2A%28S-2%29%2A%28S+%2B+2%29+=+10%2A%28S%2B2%29+%2B+10%2A%28S+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that you can simplify this a little by dividing both sides (all terms) by 2 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28S-2%29%2A%28S+%2B+2%29+=+5%2A%28S%2B2%29+%2B+5%2A%28S+-+2%29 ALIGN=MIDDLE  ALT=\"%28S-2%29%2A%28S+%2B+2%29+=+5%2A%28S%2B2%29+%2B+5%2A%28S+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now do the multiplications. First, the left side multiplies out to give:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5E2+-+4+=+5%2A%28S%2B2%29+%2B+5%2A%28S+-+2%29 ALIGN=MIDDLE  ALT=\"S%5E2+-+4+=+5%2A%28S%2B2%29+%2B+5%2A%28S+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now multiply out the two terms on the right side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5E2+-+4+=+5S+%2B+10+%2B+5S+-+10 ALIGN=MIDDLE  ALT=\"S%5E2+-+4+=+5S+%2B+10+%2B+5S+-+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combine terms on the right side. Note the +10 and -10 cancel out and you are left with the<BR>\n" );
document.write( "equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5E2+-+4+=+10S ALIGN=MIDDLE  ALT=\"S%5E2+-+4+=+10S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get this into standard quadratic form by subtracting 10S from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5E2+-+10S+-+4+=+0 ALIGN=MIDDLE  ALT=\"S%5E2+-+10S+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The statement in the problem that tells you to round off to the nearest tenth suggests that<BR>\n" );
document.write( "this problem does not have an integer solution. So let's skip trying to factor the equation<BR>\n" );
document.write( "and go directly to using the quadratic formula. The quadratic formula says that for an equation<BR>\n" );
document.write( "of the standard form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=aS%5E2+%2B+bS+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"aS%5E2+%2B+bS+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The solutions will be given by the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"S+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By comparing the equation we are trying to solve with the standard form we can see that<BR>\n" );
document.write( "a = 1, b = -10, and c = -4. Plug these values into the appropriate places in the solutions equation<BR>\n" );
document.write( "and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%2A1%2A%28-4%29+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"S+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%2A1%2A%28-4%29+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The terms inside the radical become <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100+%2B+16+=+116 ALIGN=MIDDLE  ALT=\"100+%2B+16+=+116\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute that and the equation<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28-%28-10%29+%2B-+sqrt%28+116+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"S+=+%28-%28-10%29+%2B-+sqrt%28+116+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that the denominator simplifies to 2 and the -(-10) is +10. Substituting these values<BR>\n" );
document.write( "results in the further simplified form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%2810+%2B-+sqrt%28+116+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"S+=+%2810+%2B-+sqrt%28+116+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Calculator time. The square root of 116 is 10.77032961. Substitute that value and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%2810+%2B-+10.77032961%29%2F2 ALIGN=MIDDLE  ALT=\"S+=+%2810+%2B-+10.77032961%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Look carefully at the numerator. If you use the minus sign the resulting numerator will<BR>\n" );
document.write( "be a negative value ... and it doesn't make sense to have a negative speed. So let's just<BR>\n" );
document.write( "go with the positive sign and this makes the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%2810+%2B10.77032961%29%2F2=+20.77032961%2F2+=+10.38516481 ALIGN=MIDDLE  ALT=\"S+=+%2810+%2B10.77032961%29%2F2=+20.77032961%2F2+=+10.38516481\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This means that the speed of the boat in still water is (rounding to the nearest tenth) 10.4<BR>\n" );
document.write( "miles per hour.<BR>\n" );
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document.write( "Check ... in going upstream the time is equal to distance divided by speed and that is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+10%2F8.4+=+1.19 ALIGN=MIDDLE  ALT=\"T+=+10%2F8.4+=+1.19\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours.<BR>\n" );
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document.write( "in going downstream the time is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+10%2F12.4+=+0.81 ALIGN=MIDDLE  ALT=\"T+=+10%2F12.4+=+0.81\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours.<BR>\n" );
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document.write( "So the total round trip time is 1.19 + 0.81 = 2 hours ... just as it should be. The answer<BR>\n" );
document.write( "checks so we can say that the speed of the boat in still water is 10.4 miles per hour.<BR>\n" );
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document.write( "Hope this helps you understand the problem and how you can get the answer.<BR>\n" );
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