document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=277017>Question 277017</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Because of an anticipated heavy rainstorm, the water level in a reservoir must be lowered by 1 foot. Opening Spillway A lowers the level by this amount in 4 hours, whereas opening the smaller spillway B does the job in 6 hours. How long will it take to lower the water level by 1 foot if both spillways are opening? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #201782 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=201782\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> rate * time = units \r<BR>\n" );
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document.write( "the number of units is equal to 1 (lowering the level in the reservoir by 1 foot)\r<BR>\n" );
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document.write( "Spillway A can do the job in 4 hours.\r<BR>\n" );
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document.write( "The rate at which spillway A can do the job is equal to 1/4 which means that spillway A can lower 1/4 of a foot per hour.\r<BR>\n" );
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document.write( "Spillway B does the job in 6 hours.\r<BR>\n" );
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document.write( "The rate at which spillway B can do the job is equal to 1/6 which means that spillway A can lower 1/6 of a foot per hour.\r<BR>\n" );
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document.write( "If you open both spillways, then their rates are additive.\r<BR>\n" );
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document.write( "This means that:\r<BR>\n" );
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document.write( "(1/4 + 1/6) * T = 1 where T is the number of hours it will take when both work together.\r<BR>\n" );
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document.write( "Remove parentheses to get:\r<BR>\n" );
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document.write( "(1/4)*T + (1/6)*T = 1\r<BR>\n" );
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document.write( "Multiply both sides of this equation by 24 to get:\r<BR>\n" );
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document.write( "6*T + 4*T = 24\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( "10*T = 24\r<BR>\n" );
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document.write( "Divide both sides of this equation by 10 to get:\r<BR>\n" );
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document.write( "T = 2.4\r<BR>\n" );
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document.write( "It should take 2.4 hours for both working together.\r<BR>\n" );
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document.write( "Plug into original equation to see if this is accurate.\r<BR>\n" );
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document.write( "(1/4)*2.4 + (1/6)*2.4 = 1.\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( ".6 + .4 = 1 which is true confirming the values are good.\r<BR>\n" );
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document.write( "In the following graph, y represents the rate, and x represents the time.\r<BR>\n" );
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document.write( "You can see that\r<BR>\n" );
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document.write( "when x = 4, y = 1/4 which is the rate of spillway A.<BR>\n" );
document.write( "when x = 6, y = 1/6 which is the rate of spillway B.<BR>\n" );
document.write( "when x = 2.4, y = 10/24 which is the combined rate of spillway A and B.\r<BR>\n" );
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document.write( "1/4 + 1/6 = 10/24.\r<BR>\n" );
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document.write( "The horizontal lines on the graph are 10/24, 1/4, and 1/6 respectively going from top to bottom.\r<BR>\n" );
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document.write( "They correspond to the x values of 2.4, 4, and 6 respectively going from left to right.\r<BR>\n" );
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document.write( "When spillway A and spillway B work together, they can lower the level by 1 foot in 2.4 hours.\r<BR>\n" );
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document.write( "Since Rate * Time = Work (expressed in units produced), then:\r<BR>\n" );
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document.write( "R * 2.4 = 1 which results in R = 1/2.5 which is the same as 10/24 which is the combined rate of spillway A and spillway B.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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