document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1110573>Question 1110573</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Brine containing 4 lbs/gal of salt enters a large tank at the rate of 4 gal/min and the solution well-stirred leaves at 2 gal/min. The tank initially contains 30 gal of water. • Set up an equation determining the amount of salt in the tank at any time t in minutes. • What is the amount of salt in the tank after 10 minutes?  • When will there be 170 lbs of salt in the tank? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #725808 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=725808\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> My best attempted estimate is that after 10 minutes the tank would contain<BR>\n" );
document.write( "a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128 ALIGN=MIDDLE  ALT=\"128\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lb ALIGN=MIDDLE  ALT=\"lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt (in a total of 50 gallons of solution).<BR>\n" );
document.write( "I reached that estimate through a computational approach, described below.<BR>\n" );
document.write( "The situation is complicated,<BR>\n" );
document.write( "because as brine with a fixed 4 lb/gallon concentration is entering the tank,<BR>\n" );
document.write( "dilute salt solution of varying concentration is continuously leaving the tank.<BR>\n" );
document.write( "That makes me think that the problem would require solving complicated differential equations.<BR>\n" );
document.write( "It also requires ignoring chemistry inconsistencies, and considering negligible<BR>\n" );
document.write( "all differences between the final volume of a mix of solutions and the sum of the volumes mixed.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "Ignoring all practical aspect flaws in the design of this problem,<BR>\n" );
document.write( "the concentration of salt in the tank would approach 4 lb/gallon over time,<BR>\n" );
document.write( "but it would reach exactly that value at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=time ALIGN=MIDDLE  ALT=\"time\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "As the solution is \"well stirred\", the concentration of salt in the tank, and in the solution leaving the tank would be increasing over time.<BR>\n" );
document.write( "We can simplify the problem by ignoring some of the salt losses to get a slight overestimate of the amount of salt in the tank.<BR>\n" );
document.write( " <BR>\n" );
document.write( "During the first 0.5 minutes, volume has increased by 1 gallon (2 gallons in, 1 gallon out).<BR>\n" );
document.write( "So after 0.5 minutes, the volume is 31 gallons.<BR>\n" );
document.write( "In the 2 gallons of brine that entered the tank, 8 lb of salt came in.<BR>\n" );
document.write( "As the solution is \"well stirred\", the concentration of salt in the tank,<BR>\n" );
document.write( "and the amount of salt leaving the tank during the first 0.5 minutes is not exactly zero.<BR>\n" );
document.write( "to simplify the problem, we can say that it was about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, we can say that at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=+%220.5+minutes%22 ALIGN=MIDDLE  ALT=\"t=+%220.5+minutes%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > the concentration of salt in the tank is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8lb%2F%2231+gal%22=%220.258+lb+%2F+gal%22 ALIGN=MIDDLE  ALT=\"8lb%2F%2231+gal%22=%220.258+lb+%2F+gal%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "To continue simplifying, let's say that the concentration in the stream leaving the tank<BR>\n" );
document.write( "is that same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%220.258+lb+%2F+gal%22 ALIGN=MIDDLE  ALT=\"%220.258+lb+%2F+gal%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the next 0.5 minutes.<BR>\n" );
document.write( "Then at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%221.0+minutes%22 ALIGN=MIDDLE  ALT=\"t=%221.0+minutes%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >, having gained another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lb ALIGN=MIDDLE  ALT=\"lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt,<BR>\n" );
document.write( "and having lost <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281gallon%29%2A%28%220.258+lb+%2F+gal%22%29=0.258lb ALIGN=MIDDLE  ALT=\"%281gallon%29%2A%28%220.258+lb+%2F+gal%22%29=0.258lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt,<BR>\n" );
document.write( "the tank contains <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8+lb%2B8+lb-0.258lb=15.742lb ALIGN=MIDDLE  ALT=\"8+lb%2B8+lb-0.258lb=15.742lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt.<BR>\n" );
document.write( "At the same time, the volume is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32gallons ALIGN=MIDDLE  ALT=\"32gallons\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1.0+minute ALIGN=MIDDLE  ALT=\"t=1.0+minute\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the concentration in the tank is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15.742lb%2F%2232+gal%22=%220.492+lb+%2F+gal%22 ALIGN=MIDDLE  ALT=\"15.742lb%2F%2232+gal%22=%220.492+lb+%2F+gal%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If we keep simplifying the calculation by saying that<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%220.492+lb+%2F+gal%22 ALIGN=MIDDLE  ALT=\"%220.492+lb+%2F+gal%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the concentration in the stream leaving the tank for the next 0.5 minutes,<BR>\n" );
document.write( "by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%221.5+minutes%22 ALIGN=MIDDLE  ALT=\"t=%221.5+minutes%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > the tank has<BR>\n" );
document.write( "gained another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lb ALIGN=MIDDLE  ALT=\"lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt,<BR>\n" );
document.write( "and lost <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.492lb ALIGN=MIDDLE  ALT=\"0.492lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt.<BR>\n" );
document.write( "So, at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1.5 ALIGN=MIDDLE  ALT=\"t=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes the tank contains<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15.742lb%2B8lb-0.492lb=23.25lb ALIGN=MIDDLE  ALT=\"15.742lb%2B8lb-0.492lb=23.25lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt,<BR>\n" );
document.write( "in a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33+gallons ALIGN=MIDDLE  ALT=\"33+gallons\"   DISABLED_style= \"max-width:90%; height:auto;\" > of solution,<BR>\n" );
document.write( "with a concentration of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32.35lb%2F%2233+gal%22=%220.705+lb+%2F+gal%22 ALIGN=MIDDLE  ALT=\"32.35lb%2F%2233+gal%22=%220.705+lb+%2F+gal%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Continuing with this simplified calculation, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=129lb ALIGN=MIDDLE  ALT=\"129lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10.0minutes ALIGN=MIDDLE  ALT=\"t=10.0minutes\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is an overestimate, because we are underestimating the loss of salt duringt every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.5-minute ALIGN=MIDDLE  ALT=\"0.5-minute\"   DISABLED_style= \"max-width:90%; height:auto;\" > interval.<BR>\n" );
document.write( "Reducing the calculation intervals, increases accuracy (and calculation work).<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using a similar calculation with shorter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.25-minute ALIGN=MIDDLE  ALT=\"0.25-minute\"   DISABLED_style= \"max-width:90%; height:auto;\" > intervals<BR>\n" );
document.write( "yields <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128lb ALIGN=MIDDLE  ALT=\"128lb\"   DISABLED_style= \"max-width:90%; height:auto;\" > of salt in the tank after 10 minutes.<BR>\n" );
document.write( "That is also slight overestimate, but good enough.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTES: <BR>\n" );
document.write( "I do not know what the person(s) who proposed the problem expect(s),<BR>\n" );
document.write( "but the situation is nowhere as simple as they expect.<BR>\n" );
document.write( "A problem similar to this one may appear in a chemical engineering course.<BR>\n" );
document.write( "The concentration flowing into the tank would be less, as 4 lb/US gallon (and even 4 lb/Imperial gallon) exceeds the solubility of salt in water.<BR>\n" );
document.write( "The students would know that as you mix solutions the weights of solution and salt add up,<BR>\n" );
document.write( "but the volumes do not exactly add up, so they would use data tables<BR>\n" );
document.write( "showing density of solutions as function of concentration and temperature,<BR>\n" );
document.write( "and of course, the temperatures of the tank contents and stream flowing into the tank would be stated in the problem. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );