document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=828298>Question 828298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Boating:  The table gives the amount of y gas needed to travel a fixed distance if a boat gets x miles per gallon when driven at a constant speed.\r<BR>\n" );
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document.write( "Mileage (miles per gallon), x = 11   Gas (gallons) y =7.5<BR>\n" );
document.write( "Mileage (miles per gallon), x = 13.75   Gas (gallons) y =6<BR>\n" );
document.write( "Mileage (miles per gallon), x = 15   Gas (gallons) y =5.5<BR>\n" );
document.write( "Mileage (miles per gallon), x = 16.5   Gas (gallons) y =5<BR>\n" );
document.write( "Mileage (miles per gallon), x = 27.5  Gas (gallons) y =3<BR>\n" );
document.write( "Mileage (miles per gallon), x = 33  Gas (gallons) y =?\r<BR>\n" );
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document.write( "I am not sure how to figure this out.   For the last part x=33 y=?  I guessed that y would be 2.5.  I am not sure how I arrived at that answer.  I looked a a pattern with the Ys.  If it was x was a whole number, the y would be a decimal.  If Y as a decimal then y was a whole number.  The Ys were in descending order.\r<BR>\n" );
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document.write( "Please explain how to do that answer.\r<BR>\n" );
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document.write( "Thank you!\r<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 #499222 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=499222\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> AS A MATH CLASS PROBLEM:<BR>\n" );
document.write( "If this is a problem for a math class in school, you are probably expected to do it a certain way, following a certain procedure, and/or using a certain formula, rather than using your brain. That kind of \"educational malpractice\" causes many brains to atrophy, and convinces many people that doing well in math (and physics) is just a question of memorizing a lot of facts and formulas. So wrong, so sad!<BR>\n" );
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document.write( "You are probably expected to think of this as an \"inverse' relation, as in<BR>\n" );
document.write( "\"for a certain fixed distance, the number of gallons of fuel consumed is inversely proportional to the mileage (miles per gallon) of the vehicle\"<BR>\n" );
document.write( "The more economic your vehicle (yielding more miles per gallon of fuel), the less fuel you will need to go from home to work, or from home to school.<BR>\n" );
document.write( "If you use a small car that gives you twice as many miles per gallon as your neighbors big pick up truck, you will use half the amount of fuel your neighbor uses.<BR>\n" );
document.write( "You probably were given the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%2Fx ALIGN=MIDDLE  ALT=\"y=k%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > for when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > is inversely proportional to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You were probably told that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a constant,<BR>\n" );
document.write( "and that if you know one (x,y) pair in that relationship, you can find all the others.<BR>\n" );
document.write( "Take, for example,<BR>\n" );
document.write( "Mileage (miles per gallon), x = 15 Gas (gallons) y =5.5<BR>\n" );
document.write( "Substituting those values into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%2Fx ALIGN=MIDDLE  ALT=\"y=k%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.5=k%2F5 ALIGN=MIDDLE  ALT=\"5.5=k%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > as an equation to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2F15=5.5 ALIGN=MIDDLE  ALT=\"k%2F15=5.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=5.5%2A15 ALIGN=MIDDLE  ALT=\"k=5.5%2A15\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28k=82.5%29 ALIGN=MIDDLE  ALT=\"highlight%28k=82.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now you have a relation (and more specifically, a function) to calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=82.5%2Fx ALIGN=MIDDLE  ALT=\"y=82.5%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=33 ALIGN=MIDDLE  ALT=\"x=33\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=82.5%2F33 ALIGN=MIDDLE  ALT=\"y=82.5%2F33\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=2.5%29 ALIGN=MIDDLE  ALT=\"highlight%28y=2.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Indeed, you can build the whole table with that \"formula\", which makes me wonder why they gave you so many lines in that table.<BR>\n" );
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document.write( "AS AN SAT PROBLEM:<BR>\n" );
document.write( "In this case, you are expected to use your brain, and forget formulas.<BR>\n" );
document.write( "Think as a fifth grader should be able to think.<BR>\n" );
document.write( "If the guy with the inefficient boat doing just x=16.5 miles per gallon needs y=5 gallons to go a certain distance,<BR>\n" );
document.write( "your twice as efficient boat, doing twice as many miles per gallons (x=33), will need half as much fuel:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5gal%2F2=highlight%282.5gal%29 ALIGN=MIDDLE  ALT=\"y=5gal%2F2=highlight%282.5gal%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "IF YOU HAD JUST A TABLE of x and y values (maybe data collected from a science lab experiment), you would want to plot your data, and would get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C40%2C-2%2C8%2C%0D%0Agrid%281%29%2C%0D%0Ared%28circle%2811%2C7.5%2C1%29%29%2Cred%28circle%2813.75%2C6%2C1%29%29%2C%0D%0Ared%28circle%2815%2C5.5%2C1%29%29%2Cred%28circle%2816.5%2C5%2C1%29%29%2C%0D%0Ared%28circle%2827.5%2C3%2C1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Excel, or any statistical software package would let you make that graph and calculate best fit curves for several models (theories on whether y varies linearly with x, or with x squared, or it is an exponential function, or some other relation). It will tell you that the linear model does not fit very well,<BR>\n" );
document.write( "but you can see that the points do not fit a straight line. They suggest a curve. If the statistic program had an inversely proportional model, it would tell you that it fits the data perfectly.<BR>\n" );
document.write( "Otherwise, if you were in calculus, or pre-calculus class you might say the plotted points make the relation look like a hyperbola, with the x- and y-axes as aymptotes.<BR>\n" );
document.write( "So you would try to plot <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > against <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then you would find an absolutely perfect linear fit, and would realize that real data cannot be that perfect. Your lab partner must have fabricated the data. </SPAN>\n" );
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