document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186677>Question 1186677</A>: <I><SPAN STYLE=\"word-break: break-all;\">  A very large tank initially contains 100L of pure water. Starting at time t=0 a solution with a salt concentration of 0.5kg/L is added at a rate of 6L/min. The solution is kept thoroughly mixed and is drained from the tank at a rate of 4L/min. Answer the following questions.\r<BR>\n" );
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document.write( "1. Let y(t) be the amount of salt (in kilograms) in the tank after t minutes. What differential equation does y satisfy? Use the variable y for y(t).<BR>\n" );
document.write( "Answer (in kilograms per minute): dydt= <BR>\n" );
document.write( "2. How much salt is in the tank after 20 minutes?<BR>\n" );
document.write( "Answer (in kilograms): <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 #849378 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849378\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this mixing problem:\r<BR>\n" );
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document.write( "**1. Differential Equation:**\r<BR>\n" );
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document.write( "The rate of change of salt in the tank (dy/dt) is determined by the difference between the rate of salt entering and the rate of salt leaving.\r<BR>\n" );
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document.write( "*   **Rate in:** The solution enters at 6 L/min with a concentration of 0.5 kg/L. So, the rate of salt entering is 6 L/min * 0.5 kg/L = 3 kg/min.<BR>\n" );
document.write( "*   **Rate out:** The solution leaves at 4 L/min. The concentration of salt in the tank at time *t* is y(t) / (100 + (6-4)t) = y(t) / (100 + 2t). So, the rate of salt leaving is 4 L/min * [y(t) / (100 + 2t)] kg/L = (4y) / (100 + 2t) kg/min.\r<BR>\n" );
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document.write( "Therefore, the differential equation is:\r<BR>\n" );
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document.write( "dy/dt = 3 - (4y) / (100 + 2t)\r<BR>\n" );
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document.write( "**2. Amount of Salt after 20 Minutes:**\r<BR>\n" );
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document.write( "To find the amount of salt after 20 minutes, we need to solve the differential equation with the initial condition y(0) = 0 (since the tank initially contains pure water). This is a first-order linear differential equation, and it can be solved using an integrating factor.\r<BR>\n" );
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document.write( "However, a simpler approach is to use the code I provided earlier, which employs a numerical method to solve the differential equation.\r<BR>\n" );
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document.write( "Using the code, the amount of salt in the tank after 20 minutes is approximately 44.49 kg.<BR>\n" );
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