document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117642>Question 117642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> THE SPEED OF AN AIRPLANE IN STILL AIR IS 224KM/H. THE PLANE TRAVELS 733KM AGAINST THE WIND AND 1075KM WITH THE WIND IN A TOTAL TIME OF 15HR. WHAT IS THE SPEED OF THE WIND </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85801 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=85801\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <b>Please stop using ALL CAPS as it is annoying, hard to read, and is the electronic communication equivalent of SHOUTING.  If your excuse is that you cannot type very well, my response is that it is high time you learned -- take a typing course.</b>\r<BR>\n" );
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document.write( "Now, to your problem.  Let's call the wind speed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bw%5D ALIGN=MIDDLE  ALT=\"r%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The airplane then travels at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=224%2Br%5Bw%5D ALIGN=MIDDLE  ALT=\"224%2Br%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > when it is going with the wind, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=224-r%5Bw%5D ALIGN=MIDDLE  ALT=\"224-r%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > when it is going against the wind.  Let's say the time of flight for the 'with the wind' part of the trip is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  That means the time of flight while traveling against the wind must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-t ALIGN=MIDDLE  ALT=\"15-t\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The basic formula that relates distance, rate (speed), and time is <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;\" >, so let's put it all together:\r<BR>\n" );
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document.write( "Eq 1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=733=%28224-r%5Bw%5D%29%2815-t%29 ALIGN=MIDDLE  ALT=\"733=%28224-r%5Bw%5D%29%2815-t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and\r<BR>\n" );
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document.write( "Eq 2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1075=%28224%2Br%5Bw%5D%29t ALIGN=MIDDLE  ALT=\"1075=%28224%2Br%5Bw%5D%29t\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First, solve Eq 2) 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;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1075%2F%28224%2Br%5Bw%5D%29 ALIGN=MIDDLE  ALT=\"t=1075%2F%28224%2Br%5Bw%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we have an expression 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;\" > in terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bw%5D ALIGN=MIDDLE  ALT=\"r%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > that can be substituted into Eq 1), giving us an equation in a single variable that we can attempt to solve for the value that answers the question, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bw%5D ALIGN=MIDDLE  ALT=\"r%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > or wind speed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=733=%28224-r%5Bw%5D%29%2815-%281075%2F%28224%2Br%5Bw%5D%29%29%29 ALIGN=MIDDLE  ALT=\"733=%28224-r%5Bw%5D%29%2815-%281075%2F%28224%2Br%5Bw%5D%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First, get everything on one side so that one side is zero.  Since we have the variable in both the numerator and denominator, we will end up with a quadratic.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-733%2B%28224-r%5Bw%5D%29%2815-%281075%2F%28224%2Br%5Bw%5D%29%29%29=0 ALIGN=MIDDLE  ALT=\"-733%2B%28224-r%5Bw%5D%29%2815-%281075%2F%28224%2Br%5Bw%5D%29%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we need to distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=224-r%5Bw%5D ALIGN=MIDDLE  ALT=\"224-r%5Bw%5D\"   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=-733%2B15%28224-r%5Bw%5D%29-%28%281075%28224-r%5Bw%5D%29%29%2F%28224%2Br%5Bw%5D%29%29=0 ALIGN=MIDDLE  ALT=\"-733%2B15%28224-r%5Bw%5D%29-%28%281075%28224-r%5Bw%5D%29%29%2F%28224%2Br%5Bw%5D%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Our lowest common denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=224%2Br%5Bw%5D ALIGN=MIDDLE  ALT=\"224%2Br%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-733%28224%2Br%5Bw%5D%29%2B15%28224-r%5Bw%5D%29%28224%2Br%5Bw%5D%29-%281075%28224-r%5Bw%5D%29%29%29%2F%28224%2Br%5Bw%5D%29=0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now distribute and collect terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-164192-733r%5Bw%5D%2B752640-15r%5Bw%5D%5E2-240800%2B1075r%5Bw%5D%29%2F%28224%2Br%5Bw%5D%29=0 ALIGN=MIDDLE   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=%2815r%5Bw%5D%5E2-342r-347648%29%2F%28224%2Br%5Bw%5D%29=0 ALIGN=MIDDLE  ALT=\"%2815r%5Bw%5D%5E2-342r-347648%29%2F%28224%2Br%5Bw%5D%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So far, we have developed a rational equation.  We should now consider whether or not we have any domain problems, i.e. are there values of the variable for which the left side rational expression is undefined?  Since we have taken the direction of the wind into account when we made the two expressions for airspeed, we can safely concern ourselves with only positive values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bw%5D ALIGN=MIDDLE  ALT=\"r%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since there are no positive values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bw%5D ALIGN=MIDDLE  ALT=\"r%5Bw%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > that would make the denominator zero, we don't need to be concerned about having to exclude a positive root of the equation.  Further, we can exclude any negative root(s) of the equation since negative speed is nonsensical in this context.\r<BR>\n" );
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document.write( "The roots of a rational are simply the root(s) of the numerator polynomial, excluding any values not in the domain of the entire rational expression.\r<BR>\n" );
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document.write( "All that is left is to solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15r%5Bw%5D%5E2-342r-347648%29=0 ALIGN=MIDDLE  ALT=\"15r%5Bw%5D%5E2-342r-347648%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2%2B-342x%2B-347648+=+0 ALIGN=MIDDLE  ALT=\"15x%5E2%2B-342x%2B-347648+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
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document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-342%29%5E2-4%2A15%2A-347648=20975844 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-342%29%5E2-4%2A15%2A-347648=20975844\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=20975844 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--342%2B-sqrt%28+20975844+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--342%2B-sqrt%28+20975844+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-342%29%2Bsqrt%28+20975844+%29%29%2F2%5C15+=+164.064643363594 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-342%29%2Bsqrt%28+20975844+%29%29%2F2%5C15+=+164.064643363594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-342%29-sqrt%28+20975844+%29%29%2F2%5C15+=+-141.264643363594 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-342%29-sqrt%28+20975844+%29%29%2F2%5C15+=+-141.264643363594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2%2B-342x%2B-347648 ALIGN=MIDDLE  ALT=\"15x%5E2%2B-342x%2B-347648\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2%2B-342x%2B-347648+=+15%28x-164.064643363594%29%2A%28x--141.264643363594%29 ALIGN=MIDDLE  ALT=\"15x%5E2%2B-342x%2B-347648+=+15%28x-164.064643363594%29%2A%28x--141.264643363594%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 164.064643363594, -141.264643363594.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+15%2Ax%5E2%2B-342%2Ax%2B-347648+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+15%2Ax%5E2%2B-342%2Ax%2B-347648+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Notice that there are two roots given, but we have already concluded that the negative value makes no sense.\r<BR>\n" );
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document.write( "Hence, the answer is that the wind speed is 164 km/hr.  Although the pluggable quadratic solver that I used gives the answer to several decimal places, it would be incorrect to express the answer with any greater precision than the least precise measurement in the problem givens.  Since every bit of given data is presented to the nearest whole number, there shouldn't be any decimal places in the answer.\r<BR>\n" );
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document.write( "Point to ponder:<BR>\n" );
document.write( "224 km/hr is roughly 140 miles per hour.  164 km/hr is roughly 100 miles per hour.  I find it highly improbable that the flight described would ever take place.  It would be extraordinarily difficult to find a pilot who was simultaneously smart enough to fly an airplane at all and stupid enough to take his 140 mph top speed aircraft up into the teeth of a 100 mph wind.\r<BR>\n" );
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document.write( "Hope this helps,<BR>\n" );
document.write( "John<BR>\n" );
document.write( " </SPAN>\n" );
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