document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=734629>Question 734629</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8,000 gallons and after 4 hours 2,000 gallons.\r<BR>\n" );
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document.write( "a. How fast is the pump removing water.\r<BR>\n" );
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document.write( "b. Find the slope intercept form of a line that models the amount of water in the pool. Interpret the slope. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #449103 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=449103\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. The first question can be answered the fifth grader way, without mentioning algebra.<BR>\n" );
document.write( "The pump removed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8000gallons-2000gallons=6000gallons ALIGN=MIDDLE  ALT=\"8000gallons-2000gallons=6000gallons\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the time between those two measurements.<BR>\n" );
document.write( "The time elapsed between those two measurements was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4hours-1+hour=3+hours ALIGN=MIDDLE  ALT=\"4hours-1+hour=3+hours\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The pump is removing water at a rate of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6000gallons%2F3hours=2000 ALIGN=MIDDLE  ALT=\"6000gallons%2F3hours=2000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gallons%2Fhour ALIGN=MIDDLE  ALT=\"gallons%2Fhour\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "b. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= volume of water in the pool (in gallons)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= time since the emptying started (in hours)<BR>\n" );
document.write( "(You could use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > for volume in the pool 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;\" > for time if you like that better).<BR>\n" );
document.write( "You need to write the linear function that models the emptying of the pool, showing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a function of <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 have two points from that line: point 1 and point 2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5B1%5D ALIGN=MIDDLE  ALT=\"P%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >(1,8000) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5B2%5D ALIGN=MIDDLE  ALT=\"P%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >(4,2000),<BR>\n" );
document.write( "meaning that for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=1 ALIGN=MIDDLE  ALT=\"x%5B1%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=8000 ALIGN=MIDDLE  ALT=\"y%5B1%5D=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=4 ALIGN=MIDDLE  ALT=\"x%5B2%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=2000 ALIGN=MIDDLE  ALT=\"y%5B2%5D=2000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Equation of the line:<BR>\n" );
document.write( "There is more than one \"form\" for the equation of a line, and an infinite number of versions of some of those forms, but the slope-intercept form is very popular and comes in just one unique version.<BR>\n" );
document.write( "The slope-intercept form looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Ax%2Bb ALIGN=MIDDLE  ALT=\"y=m%2Ax%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with the values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=slope ALIGN=MIDDLE  ALT=\"m=slope\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=y-intercept ALIGN=MIDDLE  ALT=\"b=y-intercept\"   DISABLED_style= \"max-width:90%; height:auto;\" > being constants that determine the line.<BR>\n" );
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document.write( "ONE WAY TO GET TO THE EQUATION OF THE LINE:<BR>\n" );
document.write( "If you are very good at solving systems of equations, you could use that skill to find the equation.<BR>\n" );
document.write( "You could substitute the coordinates of P[1] into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Ax%2Bb ALIGN=MIDDLE  ALT=\"y=m%2Ax%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8000=m%2A1%2Bb ALIGN=MIDDLE  ALT=\"8000=m%2A1%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bb=8000 ALIGN=MIDDLE  ALT=\"m%2Bb=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "you could substitute the coordinates of P[2] into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Ax%2Bb ALIGN=MIDDLE  ALT=\"y=m%2Ax%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000=m%2A4%2Bb ALIGN=MIDDLE  ALT=\"2000=m%2A4%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4m%2Bb=2000 ALIGN=MIDDLE  ALT=\"4m%2Bb=2000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then you could solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m%2Bb=8000%2C4m%2Bb=2000%29 ALIGN=MIDDLE  ALT=\"system%28m%2Bb=8000%2C4m%2Bb=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "ANOTHER WAY:<BR>\n" );
document.write( "Using what you just learned about linear functions, you can calculate the slope as the change in y divided by the change in x when we go from point 1 to point 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29=%282000-8000%29%2F%284-1%29=%28-600%29%2F3=-2000 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So far this is the same calculation as for part a, but involving algebra and talking about lines and slope.<BR>\n" );
document.write( " <BR>\n" );
document.write( "After finding the slope, you could use one of the points and the concept of slope (or the formula for the point-slope form of the equation to write the equation of the line in point-slope form. Then you could \"solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get to the slope intercept form.<BR>\n" );
document.write( "The definition of slope, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, using a point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >) representing any point in the line, and a known point A(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BA%5D ALIGN=MIDDLE  ALT=\"x%5BA%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BA%5D ALIGN=MIDDLE  ALT=\"y%5BA%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and the formula for a point-slope form of the equation are related to each other like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y-y%5BA%5D%29%2F%28x-x%5BA%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y-y%5BA%5D%29%2F%28x-x%5BA%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5BA%5D=m%28x-x%5BA%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5BA%5D=m%28x-x%5BA%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-2000 ALIGN=MIDDLE  ALT=\"m=-2000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and point (1,8000) we can write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8000=-2000%28x-1%29 ALIGN=MIDDLE  ALT=\"y-8000=-2000%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the point-slope form<BR>\n" );
document.write( "Then, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8000=-2000%28x-1%29 ALIGN=MIDDLE  ALT=\"y-8000=-2000%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8000=-2000x-2000 ALIGN=MIDDLE  ALT=\"y-8000=-2000x-2000\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2000x-2000%2B8000 ALIGN=MIDDLE  ALT=\"y=-2000x-2000%2B8000\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-2000x%2B10000%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-2000x%2B10000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the moment the pump was turned on to start emptying the pool, the volume of water in the pool was<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10000 ALIGN=MIDDLE  ALT=\"y=10000\"   DISABLED_style= \"max-width:90%; height:auto;\" >. There were 10,000 gallons of water in the pool. That is the y-intercept, the point where the line intercepts the y-axis.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C6%2C-1500%2C13500%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28line%280%2C10000%2C5%2C0%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "INTERPRETING THE SLOPE:<BR>\n" );
document.write( "The slope is the rate of increase of y as x increases, and it means that for every increase of 1 hour in the time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > spent emptying the pool, the volume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > of water in the pool increases by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2000 ALIGN=MIDDLE  ALT=\"-2000\"   DISABLED_style= \"max-width:90%; height:auto;\" > gallons. The negative sign for that \"increase\" in the volume shows that the volume is really decreasing. </SPAN>\n" );
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